# A universal pair of genus-two curves

A. Malmendier  Department of Mathematics and Statistics, Utah State
University, Logan, UT 84322<EMAIL_ADDRESS>and  T. Shaska
Department of Mathematics and Statistics, Oakland University, Rochester, MI
48309<EMAIL_ADDRESS>
###### Abstract.

Let $\mathfrak{p}$ be any point in the moduli space of genus-two curves
${\mathcal{M}}_{2}$ and $K$ its field of moduli. We provide a universal pair
of genus-two curves over its minimal field of definition $K[d]$ where $d^{2}$
is given in terms of $\mathfrak{p}$. Furthermore, $K$ is a field of definition
if and only if $d^{2}$ is a complete square in $K$.

###### 2010 Mathematics Subject Classification:

14H10,14H45

##  1\. Introduction

In [5] Igusa described the moduli space ${\mathcal{M}}_{2}$ of genus-two
curves over a field $k$. This is based on the invariants of the binary
sextics, already known to classical invariant theorists. Most of the Igusa’s
paper focuses on the cases when $\mbox{char }k<5$. For a much shorter approach
in the case of characteristic zero which makes use of geometric invariant
theory see [9].

In [11] Mestre introduced a method to construct the equation of a genus-two
curve $\mathcal{C}$ from a point in the moduli space. It was already known to
Clebsch and Bolza that the Weierstrass points of the genus-two curve were
given as the intersection of a conic and a cubic. Mestre determined the
equation of $\mathcal{C}$ by parametrizing the conic. However, this is only
possible when the conic has a rational point. Replacing such parametrization
in the equation of the cubic gives a degree six polynomial whose roots are the
projections of the Weierstrass points of $\mathcal{C}$. More precisely,
Mestre’s method assumed that for any particular point
$\mathfrak{p}\in{\mathcal{M}}_{2}(\mathbb{Q})$ the corresponding conic has a
rational point. When the conic has no rational point, then there is no genus-
two curve defined over $\mathbb{Q}$ (i.e., the field of moduli is not a field
of definition) and the algorithm does not give an output. Moreover, the case
of curves with automorphism group of order bigger than two was not considered.
An equation for such cases was provided later independently by Cardona and the
second author.

In this paper, for any moduli point $\mathfrak{p}\in{\mathcal{M}}_{2}(K)$,
where $K$ is the field of definition of $\mathfrak{p}$, we construct a
universal genus-two curve $\mathcal{C}$ defined over the minimal field of
definition, which we prove is $K[d]$, where $d^{2}$ is given in terms of the
Igusa invariants $I_{2},I_{4},I_{6},I_{10}$ or in terms of the Siegel modular
forms. To be precise, we will be constructing a universal pair of genus-two
curves corresponding to the two solutions $\pm d$ of the aforementioned
modular equation defining $d^{2}$.

The case when $d^{2}$ is a complete square in $K$ corresponds exactly to the
genus-two curves for which the field of moduli is a field of definition. In
contrast to Mestre’s approach our equation is valid even when the field of
moduli is not a field of definition, and it works for curves with extra
automorphisms as well. Our main result is the following:

Theorem:  For every point $\mathfrak{p}\in{\mathcal{M}}_{2}$ such that
$\mathfrak{p}\in{\mathcal{M}}_{2}(k)$, for some number field $K$, there is
pair of genus-two curves
$\mathcal{C}^{\pm}:y^{2}=\sum_{i=0}^{6}a_{i}^{\pm}\,x^{i},$ corresponding to
$\mathfrak{p}$, such that $a_{i}^{\pm}\in K[d]$, $i=0,\dots,6$ as given
explicitly in Equation (42). Moreover, $K[d]$ is the minimal field of
definition of $\mathfrak{p}$.

The paper is organized as follows. In Section 2 we give a brief summary of
Siegel modular forms, classical invariants of binary sextics and the relations
among them, and Igusa functions. While this material can be found in many
places in the literature, there is plenty of confusion on the labeling of such
invariants and relations among them. We also provide a short description of
the Mestre’s method and the results that we need in the following sections.

In Section 3 we give a description of the moduli space of curves via the
Siegel modular forms. In Section 4 we determine the universal curve of genus
two. We chose to express the coefficients of this curve in terms of the
Clebsch invariants $A,B,C,D$. However, it is straight forward to convert our
results to the global Igusa functions $j_{1},j_{2},j_{3}$ and the Siegel
modular forms.

##  2\. Preliminaries

###  2.1. The Siegel modular three-fold

The Siegel three-fold is a quasi-projective variety of dimension $3$ obtained
from the Siegel upper half-plane of degree two which by definition is the set
of two-by-two symmetric matrices over $\mathbb{C}$ whose imaginary part is
positive definite, i.e.,

(1)
$\mathbb{H}_{2}=\left.\left\\{\underline{\tau}=\left(\begin{array}[]{cc}\tau_{1}&z\\\
z&\tau_{2}\end{array}\right)\right|\tau_{1},\tau_{2},z\in\mathbb{C}\,,\;\textnormal{Im}\;\;\\!\\!\\!{(\tau_{1})}\,\textnormal{Im}\;\;\\!\\!\\!{(\tau_{2}})>\textnormal{Im}\;\;\\!\\!\\!{(z)}^{2}\,,\;\textnormal{Im}\;\;\\!\\!\\!{(\tau_{2})}>0\right\\}\;,$

quotiented out by the action of the modular transformations
$\Gamma_{2}:=\operatorname{Sp}_{4}(\mathbb{Z})$, i.e.,

(2) $\mathcal{A}_{2}=\mathbb{H}_{2}/\Gamma_{2}\;.$

Each $\underline{\tau}\in\mathbb{H}_{2}$ determines a principally polarized
complex abelian surface

$\mathbf{A}_{\underline{\,\tau}}=\mathbb{C}^{2}/\langle\mathbb{Z}^{2}\oplus\underline{\tau}\,\mathbb{Z}^{2}\rangle$

with period matrix
$(\underline{\tau},\mathbb{I}_{2})\in\mathrm{Mat}(2,4;\mathbb{C})$. Two
abelian surfaces $\mathbf{A}_{\underline{\,\tau}}$ and
$\mathbf{A}_{\underline{\,\tau}^{\prime}}$ are isomorphic if and only if there
is a symplectic matrix

(3) $M=\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right)\in\Gamma_{2}$

such that
$\underline{\tau}^{\prime}=M(\underline{\tau}):=(A\underline{\tau}+B)(C\underline{\tau}+D)^{-1}$.
It follows that the Siegel three-fold $\mathcal{A}_{2}$ is also the set of
isomorphism classes of principally polarized abelian surfaces. The sets of
abelian surfaces that have the same endomorphism ring form sub-varieties of
$\mathcal{A}_{2}$. The endomorphism ring of principally polarized abelian
surface tensored with $\mathbb{Q}$ is either a quartic CM field, an indefinite
quaternion algebra, a real quadratic field or in the generic case
$\mathbb{Q}$. Irreducible components of the corresponding subsets in
$\mathcal{A}_{2}$ have dimensions $0,1,2$ and are known as CM points, Shimura
curves and Humbert surfaces, respectively.

The Humbert surface $H_{\Delta}$ with invariant $\Delta$ is the space of
principally polarized abelian surfaces admitting a symmetric endomorphism with
discriminant $\Delta$. It turns out that $\Delta$ is a positive integer
$\equiv 0,1\mod 4$. In fact, $H_{\Delta}$ is the image inside
$\mathcal{A}_{2}$ under the projection of the rational divisor associated to
the equation

(4) $a\,\tau_{1}+b\,z+c\,\tau_{3}+d\,(z^{2}-\tau_{1}\,\tau_{2})+e=0\;,$

with integers $a,b,c,d,e$ satisfying $\Delta=b^{2}-4\,a\,c-4\,d\,e$ and
$\underline{\tau}=\bigl{(}\begin{smallmatrix}\tau_{1}&z\\\
z&\tau_{2}\end{smallmatrix}\bigr{)}\in\mathbb{H}_{2}$. For example, inside of
$\mathcal{A}_{2}$ sit the Humbert surfaces $H_{1}$ and $H_{4}$ that are
defined as the images under the projection of the rational divisor associated
to $z=0$ and $\tau_{1}=\tau_{2}$, respectively. In fact, the singular locus of
$\mathcal{A}_{2}$ has $H_{1}$ and $H_{4}$ as its two connected components. As
analytic spaces, the surfaces $H_{1}$ and $H_{4}$ are each isomorphic to the
Hilbert modular surface

(5)
$\Big{(}(\operatorname{SL}_{2}(\mathbb{Z})\times\operatorname{SL}_{2}(\mathbb{Z}))\rtimes\mathbb{Z}_{2}\Big{)}\backslash\Big{(}\mathbb{H}\times\mathbb{H}\Big{)}\;.$

For a more detailed introduction to Siegel modular form, Humbert surfaces, and
the Satake compactification of the Siegel modular threefold we refer to
Freitag’s book [3].

###  2.2. Siegel modular forms

In general, we can define the Eisenstein series $\psi_{2k}$ of degree $g$ and
weight $2k$ (where we assume $2k>g+1$ for convergence) by setting

(6) $\psi_{2k}(\underline{\tau})=\sum_{(C,D)}\mbox{det
}(C\cdot\underline{\tau}+D)^{-2k}\;,$

where the sum runs over non-associated bottom rows $(C,D)$ of elements in
$\operatorname{Sp}_{2g}(\mathbb{Z})$ where non-associated means with respect
to the multiplication by $\mathrm{GL}(g,\mathbb{Z})$. In the following, we
will always assume $g=2$ in the definition of $\psi_{2k}$. Using Igusa’s
definition [6, Sec. 8, p. 195] we define a cusp form of weight $10$ by

(7)
$\begin{split}\chi_{10}(\underline{\tau})&=-\frac{43867}{2^{12}\,3^{5}\,5^{2}\,7\cdot
53}\left(\psi_{4}(\underline{\tau})\,\psi_{6}(\underline{\tau})-\psi_{10}(\underline{\tau})\right)\;.\end{split}$

It is well known that the vanishing divisor of the cusp form $\chi_{10}$ is
the Humbert surface $H_{1}$ because a period point $\underline{\tau}$ is
equivalent to a point with $z=0$ if and only if
$\chi_{10}\big{(}\underline{\tau}\big{)}=0$. Based on Igusa’s definition [6,
Sec. 8, p. 195] and the work in [10] we define a second cusp form
$\chi_{12}$ of weight $12$ by

(8) $\begin{split}\chi_{12}(\underline{\tau})&=\frac{131\cdot
593}{2^{13}\,3^{7}\,5^{3}\,7^{2}\,337}\left(3^{2}\,7^{2}\,\psi_{4}^{3}(\underline{\tau})+2\cdot
5^{3}\,\psi_{6}^{2}(\underline{\tau})-691\,\psi_{12}(\underline{\tau})\right)\;.\end{split}$

Igusa proved [7, 8] that the ring of Siegel modular forms is generated by
$\psi_{4}$, $\psi_{6}$, $\chi_{10}$, $\chi_{12}$ and by one more cusp form
$\chi_{35}$ of odd weight $35$ whose square is the following polynomial [7, p.
849] in the even generators

(9)
$\begin{split}\chi_{35}^{2}&=\frac{1}{2^{12}\,3^{9}}\;\chi_{10}\,\Big{(}2^{24}\,3^{15}\;\chi_{12}^{5}-2^{13}\,3^{9}\;\psi_{4}^{3}\,\chi_{12}^{4}-2^{13}\,3^{9}\;\psi_{6}^{2}\,\chi_{12}^{4}+3^{3}\;\psi_{4}^{6}\,\chi_{12}^{3}\\\
&-2\cdot
3^{3}\;\psi_{4}^{3}\,\psi_{6}^{2}\,\chi_{12}^{3}-2^{14}\,3^{8}\;\psi_{4}^{2}\,\psi_{6}\,\chi_{10}\,\chi_{12}^{3}-2^{23}\,3^{12}\,5^{2}\,\psi_{4}\,\chi_{10}^{2}\,\chi_{12}^{3}+3^{3}\,\psi_{6}^{4}\,\chi_{12}^{3}\\\
&+2^{11}\,3^{6}\,37\,\psi_{4}^{4}\,\chi_{10}^{2}\,\chi_{12}^{2}+2^{11}\,3^{6}\,5\cdot
7\,\psi_{4}\,\psi_{6}^{2}\,\chi_{10}^{2}\,\chi_{12}^{2}-2^{23}\,3^{9}\,5^{3}\,\psi_{6}\,\chi_{10}^{3}\,\chi_{12}^{2}\\\
&-3^{2}\,\psi_{4}^{7}\,\chi_{10}^{2}\,\chi_{12}+2\cdot
3^{2}\,\psi_{4}^{4}\,\psi_{6}^{2}\,\chi_{10}^{2}\,\chi_{12}+2^{11}\,3^{5}\,5\cdot
19\,\psi_{4}^{3}\,\psi_{6}\,\chi_{10}^{3}\,\chi_{12}\\\
&+2^{20}\,3^{8}\,5^{3}\,11\,\psi_{4}^{2}\,\chi_{10}^{4}\,\chi_{12}-3^{2}\,\psi_{4}\,\psi_{6}^{4}\,\chi_{10}^{2}\,\chi_{12}+2^{11}\,3^{5}\,5^{2}\,\psi_{6}^{3}\,\chi_{10}^{3}\,\chi_{12}-2\,\psi_{4}^{6}\,\psi_{6}\,\chi_{10}^{3}\\\
&-2^{12}\,3^{4}\,\psi_{4}^{5}\,\chi_{10}^{4}+2^{2}\,\psi_{4}^{3}\,\psi_{6}^{3}\,\chi_{10}^{3}+2^{12}\,3^{4}\,5^{2}\,\psi_{4}^{2}\,\psi_{6}^{2}\,\chi_{10}^{4}+2^{21}\,3^{7}\,5^{4}\,\psi_{4}\,\psi_{6}\,\chi_{10}^{5}\\\
&-2\,\psi_{6}^{5}\,\chi_{10}^{3}+2^{32}\,3^{9}\,5^{5}\,\chi_{10}^{6}\Big{)}\;.\end{split}$

Hence, the expression $Q:=2^{12}\,3^{9}\,\chi_{35}^{2}/\chi_{10}$ is a
polynomial of degree $60$ in the even generators. The following fact is well-
known [4]:

######  Proposition 1.

The vanishing divisor of $Q$ is the Humbert surface $H_{4}$, i.e., a period
point $\underline{\tau}$ is equivalent to a point with $\tau_{1}=\tau_{2}$ if
and only if $Q\big{(}\underline{\tau}\big{)}=0$. Accordingly, the vanishing
divisor of $\chi_{35}$ is the formal sum $H_{1}+H_{4}$ of Humbert surfaces,
that constitutes the singular locus of $\mathcal{A}_{2}$.

In accordance with Igusa [6, Thm. 3] we also introduce the following ratios
of Siegel modular forms

(10)
$\mathbf{x}_{1}=\dfrac{\psi_{4}\,\chi_{10}^{2}}{\chi_{12}^{2}},\quad\mathbf{x}_{2}=\dfrac{\psi_{6}\,\chi_{10}^{3}}{\chi_{12}^{3}},\quad\mathbf{x}_{3}=\dfrac{\chi_{10}^{6}}{\chi_{12}^{5}}\;,$

as well as

(11)
$\mathbf{y}_{1}=\dfrac{\mathbf{x}_{1}^{3}}{\mathbf{x}_{3}}=\dfrac{\psi_{4}^{3}}{\chi_{12}},\quad\mathbf{y}_{2}=\dfrac{\mathbf{x}_{2}^{2}}{\mathbf{x}_{3}}=\dfrac{\psi_{6}^{2}}{\chi_{12}},\quad\mathbf{y}_{3}=\dfrac{\mathbf{x}_{1}^{2}\,\mathbf{x}_{2}}{\mathbf{x}_{3}}=\dfrac{\psi_{4}^{2}\,\psi_{6}\,\chi_{10}}{\chi_{12}}\;,$

where we have suppressed the dependence of each Siegel modular form on
$\underline{\tau}$. These ratios have the following asymptotic expansion as
$z\to 0$ [6, pp. 180–182] in terms of ordinary Eisenstein series

(12) $\begin{split}\mathbf{x}_{1}&=E_{4}(\tau_{1})\;E_{4}(\tau_{2})\,(\pi
z)^{4}+O(z^{5})\;,\\\ \mathbf{x}_{2}&=E_{6}(\tau_{1})\;E_{6}(\tau_{2})\,(\pi
z)^{6}+O(z^{7})\;,\\\
\mathbf{x}_{3}&=\eta^{24}(\tau_{1})\;\eta^{24}(\tau_{2})\,(\pi
z)^{12}+O(z^{13})\;,\\\ \end{split}$

and

(13) $\begin{split}\mathbf{y}_{1}&=j(\tau_{1})\,j(\tau_{2})+O(z^{2})\;,\\\
\mathbf{y}_{2}&=\Big{(}1728-j(\tau_{1})\Big{)}\,\Big{(}1728-j(\tau_{2})\Big{)}+O(z^{2})\;,\\\
\mathbf{y}_{3}&=\dfrac{E_{4}^{2}(\tau_{1})\,E_{4}^{2}(\tau_{2})\,E_{6}(\tau_{1})\,E_{6}(\tau_{2})}{\eta^{24}(\tau_{1})\,\eta^{24}(\tau_{2})}\,(\pi
z)^{2}+O(z^{3})\;,\end{split}$

where we have set

(14)
$\begin{split}j(\tau_{j})&=\dfrac{1728\,E_{4}^{3}(\tau_{j})}{E_{4}^{3}(\tau_{j})-E_{6}^{2}(\tau_{j})}=\dfrac{E_{4}^{3}(\tau_{j})}{\eta^{24}(\tau_{j})}\;,\\\
1728-j(\tau_{j})&=\dfrac{1728\,E_{6}^{2}(\tau_{j})}{E_{4}^{3}(\tau_{j})-E_{6}^{2}(\tau_{j})}=\dfrac{E_{6}^{2}(\tau_{j})}{\eta^{24}(\tau_{j})}\;.\end{split}$

###  2.3. Sextics and Igusa invariants

We write the equation defining a genus-two curve $\mathcal{C}$ by a degree-six
polynomial or sextic in the form

(15)
$\mathcal{C}:\;y^{2}=f(x)=a_{0}\,\prod_{i=1}^{6}(x-\alpha_{i})=\sum_{i=0}^{6}a_{i}\,x^{6-i}\;.$

The roots $\\{\alpha_{i}\\}_{i=1}^{6}$ of the sextic are the six ramification
points of the map $\mathcal{C}\to\mathbb{P}^{1}$. Their pre-images on
$\mathcal{C}$ are the six Weierstrass points. The isomorphism class of $f$
consists of all equivalent sextics where two sextics are considered equivalent
if there is a linear transformation in $\mathrm{GL}(2,\mathbb{C})$ which takes
the set of roots to the roots of the other.

The ring of invariants of binary sextics is generated by the Igusa invariants
$(I_{2},I_{4},I_{6},I_{10})$ as defined in [9, Eq. (9)], which are the same
invariants as the ones denoted by
$(A^{\prime},B^{\prime},C^{\prime},D^{\prime})$ in [11, p. 319] and also the
same invariants as $(A,B,C,D)$ in [6, p. 176]. For expressions of such
invariants in terms of the coefficients $a_{0},\dots,a_{6}$ of the binary
sextic, or $I_{k}\in\mathbb{Z}[a_{0},\dots,a_{6}]$ for $k\in\\{2,4,6,10\\}$;
see [9, Eq. (11)].

Thus, the invariants of a sextic define a point in a weighted projective space
$[I_{2}:I_{4}:I_{6}:I_{10}]\in\mathbb{WP}^{3}_{(2,4,6,10)}$. It was shown in
[6] that points in the projective variety
$\operatorname{Proj}\,\mathbb{C}[I_{2},I_{4},I_{6},I_{10}]$ which are not on
$I_{10}=0$ form the variety $\mathcal{U}_{6}$ of moduli of sextics.
Equivalently, points in the weighted projective space
$\\{[I_{2}:I_{4}:I_{6}:I_{10}]\in\mathbb{WP}^{3}_{(2,4,6,10)}:I_{10}\not=0\\}$
are in one-to-one correspondence with isomorphism classes of sextics.

Often the _Clebsch invariants_ $(A,B,C,D)$ of a sextic are used as well. They
are defined in terms of the transvectants of the binary sextics; see [2] for
details. Clebsch invariants $(A,B,C,D)$ are related to the Igusa-Clebsch
invariants by the equations

(16) $\begin{split}I_{2}=&\;-120\,A\;,\\\ I_{4}=&\;-720\,A^{2}+6750\,B\;,\\\
I_{6}=&\;\phantom{X;}8640\,A^{3}-108000\,A\,B+202500\,C\;,\\\
I_{10}=&\;-62208\,A^{5}+972000\,A^{3}\,B+1620000\,A^{2}\,C\\\
&\;-3037500\,A\,B^{2}-6075000\,B\,C-4556250\,D\;.\end{split}$

Conversely, the invariants $(A,B,C,D)$ are polynomial expressions in the Igusa
invariants $(I_{2},I_{4},I_{6},I_{10})$ with rational coefficients:

(17) $\begin{split}A=&\;-\frac{1}{2^{3}3\cdot 5}\,I_{2}\;,\\\
B=&\;\phantom{-}\frac{1}{2^{2}3^{3}5^{3}}\left(I_{2}^{2}+20\,I_{4}\right)\;,\\\
C=&\;-\frac{1}{2^{5}3^{5}5^{6}}\left(I_{2}^{3}+60\,I_{2}I_{4}-600\,I_{6}\right)\;,\\\
D=&\;-\frac{1}{2^{8}3^{9}5^{10}}\left(9\,I_{2}^{5}+700\,I_{2}^{3}I_{4}-3600\,I_{2}^{2}I_{6}\right.\\\
&\left.\qquad-12400\,I_{2}I_{4}^{2}+48000\,I_{4}\,I_{6}+10800000\,I_{10}\right)\;.\end{split}$

###  2.4. Igusa functions

The Igusa functions or absolute invariants (i.e.,
$\mathrm{GL}(2,\mathbb{C})$-invariants) are defined as

$j_{1}=\frac{I_{2}^{5}}{I_{10}},\quad
j_{2}=\frac{I_{4}I_{2}^{3}}{I_{10}},\quad j_{3}=\frac{I_{6}I_{2}^{2}}{I_{10}}$

Notice that other sets of absolute invariants have been used as well in the
literature. In this paper we will use these invariants since they are defined
everywhere on ${\mathcal{M}}_{2}$.

###  2.5. Recovering the equation of the curve from invariants

As above, let $\mathfrak{p}\in{\mathcal{M}}_{2}$ and $\mathcal{C}$ a genus-two
curve corresponding to $\mathfrak{p}$ defined by a sextic polynomial $f$.
Then, $\mbox{Aut}(\mathfrak{p})$ is a finite group as described in [12]. The
quotient space $\mathcal{C}/\mbox{Aut}(\mathfrak{p})$ is a genus zero curve
and therefore isomorphic to a conic. Since conics are in one to one
correspondence with three-by-three symmetric matrices (up to equivalence),
then let $M=\left[A_{ij}\right]$ be the symmetric matrix corresponding to this
conic.

Let $\mathbf{X}=[X_{1}:X_{2}:X_{3}]\in\mathbb{P}^{2}$ and

$\mathcal{Q}:\quad\,\,\mathbf{X}^{t}\cdot
M\cdot\mathbf{X}=\sum_{i,j=1}^{3}A_{ij}\,X_{i}X_{j}=0$

be the conic corresponding to $M$. Clebsch [2] determined the entries of this
matrix $M$ as follows

(18) $\begin{split}A_{11}&=2\,C+\frac{1}{3}\,A\,B\;,\\\
A_{22}&=A_{13}=\,D\;,\\\
A_{33}&=\frac{1}{2}\,B\,D+\frac{2}{9}\,C\,(B^{2}+A\,C)\;,\\\
A_{23}&=\frac{1}{3}\,B\,(B^{2}+A\,C)+\frac{1}{3}\,C\,(2\,C+\frac{1}{3}\,A\,B)\;,\\\
A_{12}&=\frac{2}{3}\,(B^{2}+A\,C)\;.\end{split}$

One can obtain from the sextic $f$ three binary quadrics $\mathsf{y}_{i}(x)$
with $i=1,2,3$ by an operation called ‘Überschiebung’ [11, p. 317]. The
quadrics $\mathsf{y}_{i}$ for $i=1,2,3$ have the property that their
coefficients are polynomial expressions in the coefficients of $f$ with
rational coefficients. Moreover, under the operation
$f(x)\mapsto\tilde{f}(x)=f(-x)$ the quadrics change according to
$\mathsf{y}_{i}(x)\mapsto\tilde{\mathsf{y}}_{i}(x)=\mathsf{y}_{i}(-x)$ for
$i=1,2,3$. Hence, they are not invariants of the sextic $f$. The coefficients
$A_{ij}$ in Equations (18) satisfy
$A_{ij}=(\mathsf{y}_{i}\mathsf{y}_{j})_{2}$.111For two binary forms $f,g$ of
degree $m$ and $n$, respectively, we denote the Überschiebung of order $k$ by
$(fg)_{k}=(-1)^{k}(gf)_{k}$. For $\tilde{f}(x)=f(-x)$ and $\tilde{g}(x)=g(-x)$
and $m=n=k$, we have $(fg)_{m}=(-1)^{m}(\tilde{f}\tilde{g})_{m}$. Therefore,
the coefficients $A_{ij}$ are invariant under the operation
$f(x)\mapsto\tilde{f}(x)=f(-x)$, and the locus $D=0$ is equivalent to

(19)
$D=0\quad\Leftrightarrow\quad(\mathsf{y}_{1}\mathsf{y}_{3})_{2}=(\mathsf{y}_{2}\mathsf{y}_{2})_{2}=0\;.$

We define $R$ to be $1/2$ times the determinant of the three binary quadrics
$\mathsf{y}_{i}$ for $i=1,2,3$ with respect to the basis $x^{2},x,1$. If one
extends the operation of Überschiebung by product rule [11, p. 317], then $R$
can be re-written as

(20)
$R=-(\mathsf{y}_{1}\mathsf{y}_{2})_{1}\,(\mathsf{y}_{2}\mathsf{y}_{3})_{1}\,(\mathsf{y}_{3}\mathsf{y}_{1})_{1},$

or, equivalently, as

(21)
$\begin{split}R\,=\,-\frac{1}{2}\,\Big{(}\,&\mathsf{y}_{1,yy}\,\mathsf{y}_{2,xy}\,\mathsf{y}_{3,xx}-\mathsf{y}_{1,yy}\,\mathsf{y}_{2,xx}\,\mathsf{y}_{3,xy}-\mathsf{y}_{1,xy}\,\mathsf{y}_{2,yy}\,\mathsf{y}_{3,xx}\\\
+&\,\mathsf{y}_{1,xy}\,\mathsf{y}_{2,xx}\,\mathsf{y}_{3,yy}+\mathsf{y}_{1,xx}\,\mathsf{y}_{2,yy}\,\mathsf{y}_{3,xy}-\mathsf{y}_{1,xx}\,\mathsf{y}_{2,xy}\,\mathsf{y}_{3,yy}\Big{)}\;.\end{split}$

It is then obvious that under the operation $f(x)\mapsto\tilde{f}(x)=f(-x)$
the determinant $R$ changes its sign, i.e., $R(f)\mapsto R(\tilde{f})=-R(f)$.
A straightforward calculation shows that

(22) $R^{2}=\frac{1}{2}\,\left|\begin{array}[]{ccc}A_{11}&A_{12}&A_{13}\\\
A_{12}&A_{22}&A_{23}\\\ A_{13}&A_{23}&A_{33}\end{array}\right|\;,$

where $A_{ij}$ are the Clebsch invariants in Equations (18). Like the
coefficients $A_{ij}$, $R^{2}$ is invariant under the operation
$f(x)\mapsto\tilde{f}(x)=f(-x)$ and must be a polynomial in
$(I_{2},I_{4},I_{6},I_{10})$. Bolza [1] described the possible automorphism
groups of genus-two curves defined by sextics. In particular, he provided
effective criteria for the cases when the automorphism group of the sextic
curve in Equation (15) is nontrivial. For a detailed discussion of the
automorphism groups of genus-two curve defined over any field $k$ and the
corresponding loci in ${\mathcal{M}}_{2}$ see [12].

We have the following lemma summarizing our discussion:

######  Lemma 2.

We have the following statements:

  1. (1)

$R^{2}$ is a order 30 invariant of binary sextics expressed as a polynomial in
$(I_{2},I_{4},I_{6},I_{10})$ as in [12, Eq. (17)] given by plugging Equations
(17) and (18) into Equation (22).

  2. (2)

The locus of curves $\mathfrak{p}\in{\mathcal{M}}_{2}$ such that
$V_{4}\hookrightarrow\mbox{Aut}(\mathfrak{p})$ is a two-dimensional
irreducible rational subvariety of ${\mathcal{M}}_{2}$ given by the equation
$R^{2}=0$ and a birational parametrization given by the $u,v$-invariants as in
[12, Thm. 1].

Notice that the Igusa invariants $I_{2},I_{4},I_{6},I_{10}$ are denoted by
$J_{2},\dots,J_{10}$ in [12, Eq. (17)]. From now on we will denote
$I_{30}:=R^{2}$. The relation between all aforementioned invariants and Siegel
modular forms, in particular the relation between $\chi_{35}$ and $R^{2}$,
will be given in the next section.

####  2.5.1. The cubic curve

In the previous chapter we introduced the invariant $R^{2}$ for any binary
sextic given by $f$. To the corresponding symmetric matrix $M$ with
$R^{2}=\mbox{det }{\\!M}/2$ and the coefficients
$A_{ij}=(\mathsf{y}_{i}\mathsf{y}_{j})_{2}$ of order zero invariant under the
operation $f(x)\mapsto\tilde{f}(x)=f(-x)$, we associated a conic
$\mathcal{Q}$. Similarly, there is also a cubic curve given by the equation

$\mathcal{T}:\quad\sum_{1\leq i,j,k\leq 3}a_{ijk}\,X_{i}X_{j}X_{k}=0\,,$

where the coefficients $a_{ijk}$ are of order zero and invariant under
$f(x)\mapsto\tilde{f}(x)=f(-x)$. In terms of Überschiebung the coefficients
are given by

(23)
$a_{ijk}=(f\mathsf{y}_{i})_{2}\,(f\mathsf{y}_{j})_{2}\,(f\mathsf{y}_{k})_{2}\;.$

The coefficients $a_{ijk}$ are given explicitly as follows:

(24) $\begin{split}36\,a_{111}&=8(A^{2}C-6BC+9D),\\\
36\,a_{112}&=4(2B^{3}+4ABC+12C^{2}+3AD),\\\
36\,a_{113}&=36\,a_{122}=4(AB^{3}+4/3\,A^{2}BC+4B^{2}C+6AC^{2}+3BD),\\\
36\,a_{123}&=2(2B^{4}+4AB^{2}C+4/3\,A^{2}C^{2}+4BC^{2}+3ABD+12CD),\\\
36\,a_{133}&=2\left(AB^{4}+4/3\,A^{2}B^{2}C+16/3\,B^{3}C\right.\\\
&\left.+26/3\,ABC^{2}+8C^{3}+3B^{2}D+2ACD\right),\\\
36\,a_{222}&=4(3B^{4}+6AB^{2}C+8/3\,A^{2}C^{2}+2BC^{2}-3CD),\\\
36\,a_{223}&=2(-2/3\,B^{3}C-4/3\,ABC^{2}-4C^{3}+9B^{2}D+8ACD),\\\
36\,a_{233}&=2(B^{5}+2AB^{3}C+8/9\,A^{2}BC^{2}+2/3\,B^{2}C^{2}-BCD+9D^{2}),\\\
36\,a_{333}&=-2B^{4}C-4AB^{2}C^{2}-16/9\,A^{2}C^{3}-4/3\,BC^{3}\\\
&+9B^{3}D+12ABCD+20C^{2}D.\end{split}$

The intersection of the conic $\mathcal{Q}$ with the cubic $\mathcal{T}$
consists of six points which are the zeroes of a polynomial $f(t)$ of degree 6
in the parameter $t$. The roots of this polynomial are the images of the
Weierstrass points under the hyperelliptic projection. Hence, the affine
equation of the genus-two curve corresponding to $\mathfrak{p}$ is given by
$y^{2}=f(t)$.

Mestre’s method is briefly described as follows: if the conic $\mathcal{Q}$
has a rational point over $\mathbb{Q}$, then this leads to a parametrization
of $\mathcal{Q}$, say $(h_{1}(t),h_{2}(t),h_{3}(t))$. Substitute
$X_{1},X_{2},X_{3}$ by $h_{1}(t),h_{2}(t),h_{3}(t)$ in the cubic $\mathcal{T}$
and we get the degree 6 polynomial $f(t)$. However, if the conic has no
rational point or $R^{2}=\mbox{det }M=0$ the method obviously fails. In
section 4 we determine such intersection $\mathcal{T}\cap\mathcal{Q}$ over a
quadratic extension which is always possible. Therefore, our approach works in
all cases and provides an equation for the genus-two curve over its minimal
field of definition (cf. Section 4).

##  3\. The moduli space of genus-two curves

Suppose that $\mathcal{C}$ is an irreducible projective non-singular curve. If
the self-intersection is $\mathcal{C}\cdot\mathcal{C}=2$ then $\mathcal{C}$ is
a curve of genus two. For every curve $\mathcal{C}$ of genus two there exists
a unique pair $(\mathrm{Jac}(\mathcal{C}),j_{\mathcal{C}})$ where
$\mathrm{Jac}(\mathcal{C})$ is an abelian surface, called the Jacobian variety
of the curve $\mathcal{C}$, and
$j_{\mathcal{C}}:\mathcal{C}\to\mathrm{Jac}(\mathcal{C})$ is an embedding. One
can always regain $\mathcal{C}$ from the pair
$(\mathrm{Jac}(\mathcal{C}),\mathcal{P})$ where $\mathcal{P}=[\mathcal{C}]$ is
the class of $\mathcal{C}$ in the Néron-Severi group
$\mathrm{NS}(\mathrm{Jac}(\mathcal{C}))$. Thus, if $\mathcal{C}$ is a genus-
two curve, then $\mathrm{Jac}(\mathcal{C})$ is a principally polarized abelian
surface with principal polarization $\mathcal{P}=[\mathcal{C}]$, and the map
sending a curve $\mathcal{C}$ to its Jacobian variety
$\mathrm{Jac}(\mathcal{C})$ is injective. In this way, the variety of moduli
of curves of genus two is also the moduli space of their Jacobian varieties
with canonical polarization. Furthermore, Torelli’s theorem states that the
map sending a curve $\mathcal{C}$ to its Jacobian variety
$\mathrm{Jac}(\mathcal{C})$ induces a birational map from the moduli space
$\mathcal{M}_{2}$ of genus-two curves to the complement of the Humbert surface
$H_{1}$ in $\mathcal{A}_{2}$, i.e.,
$\mathcal{A}_{2}-\textnormal{supp}{(\chi_{10})}_{0}$.

One can then ask what the Igusa-Clebsch invariants of a genus-two curve
$\mathcal{C}$ defined by a sextic curve $f$ are in terms of $\underline{\tau}$
such that $(\underline{\tau},\mathbb{I}_{2})\in\mathrm{Mat}(2,4;\mathbb{C})$
is the period matrix of the principally polarized abelian surface
$\mathbf{A}_{\underline{\tau}}=\mathrm{Jac}(\mathcal{C})$. Based on the
asymptotic behavior in Equations (12) and (13), Igusa [7, p. 848] proved that
the relations are as follows:

(25) $\begin{split}I_{2}&=-2^{3}\cdot
3\,\dfrac{\chi_{12}(\underline{\tau})}{\chi_{10}(\underline{\tau})}\;,\\\
I_{4}&=\phantom{-}2^{2}\,\psi_{4}(\underline{\tau})\;,\\\
I_{6}&=-\frac{2^{3}}{3}\,\psi_{6}(\underline{\tau})-2^{5}\,\dfrac{\psi_{4}(\underline{\tau})\,\chi_{12}(\underline{\tau})}{\chi_{10}(\underline{\tau})}\;,\\\
I_{10}&=-2^{14}\,\chi_{10}(\underline{\tau})\;.\end{split}$

Thus, we find that the point $[I_{2}:I_{4}:I_{6}:I_{10}]$ in weighted
projective space equals

(26)
$\begin{split}\Big{[}2^{3}\,3\,(3r\chi_{12})\,:\,2^{2}3^{2}\,\psi_{4}\,(r\chi_{10})^{2}\,:\,2^{3}\,3^{2}\,\Big{(}4\psi_{4}\,(3r\chi_{12})+\psi_{6}\,(r\chi_{10})\Big{)}\,(r\chi_{10})^{2}:2^{2}\,(r\chi_{10})^{6}\Big{]}\end{split}$

with $r\not=0$. Substituting (25) into Equations (16), (18) it also follows
that

(27)
$\begin{split}I_{30}=R^{2}&=-2^{6}\,3^{-27}\,5^{-20}\;\dfrac{Q\Big{(}\psi_{4}(\underline{\tau}),\psi_{6}(\underline{\tau}),\chi_{10}(\underline{\tau}),\chi_{12}(\underline{\tau})\Big{)}}{\chi_{10}(\underline{\tau})^{3}}\\\
&=-\left(2^{9}\,3^{-9}\,5^{-10}\;\dfrac{\chi_{35}(\underline{\tau})}{\chi_{10}(\underline{\tau})^{2}}\right)^{2}\;,\end{split}$

where $Q$ and $R^{2}$ where defined in Equation (9) and (22), respectively. In
terms of Igusa invariants we have

(28)
$\begin{split}&\chi_{35}^{2}=-\frac{3^{18}\,5^{20}}{2^{74}}\,I_{10}^{4}\,I_{30}.\end{split}$

For $I_{2}\not=0$ we use the variables
$\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3}$ from Equations (10) to write

(29)
$\begin{split}\Big{[}I_{2}:I_{4}:I_{6}:I_{10}\Big{]}=\left[1:\frac{1}{2^{4}\,3^{2}}\,\mathbf{x}_{1}:\frac{1}{2^{6}\,3^{4}}\,\mathbf{x}_{2}+\frac{1}{2^{4}\,3^{3}}\,\mathbf{x}_{1}:\frac{1}{2\cdot
3^{5}}\,\mathbf{x}_{3}\right]\in\mathbb{WP}^{3}_{(2,4,6,10)}\;.\end{split}$

We summarize the above discussion as follows:

######  Remark 3.

  1.   2. (1)

The modulus point $\underline{\tau}$ is equivalent to a point with
$\tau_{1}=\tau_{2}$ or $[\underline{\tau}]\in H_{4}\subset\mathcal{A}_{2}$
such that the corresponding sextic curve has an extra automorphism if and only
if $Q(\underline{\tau})=0$.

  3. (2)

The modulus point $\underline{\tau}$ is equivalent to a point with $z=0$ or
$[\underline{\tau}]\in H_{1}\subset\mathcal{A}_{2}$ such that the principally
polarized abelian surface is a product of two elliptic curves
$\mathbf{A}_{\underline{\tau}}=E_{\tau_{1}}\times E_{\tau_{2}}$ if and only if
$\chi_{10}(\underline{\tau})=0$. The elliptic modular parameters are then
determined by Equations (29) and (12).

Since the invariants $I_{4},I_{6},I_{10}$ vanish simultaneously at sextics
with triple roots all such abelian surfaces are mapped to
$[1:0:0:0]\in\mathbb{WP}^{3}_{(2,4,6,10)}$ with uniformizing affine
coordinates $\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3}$ around it. Blowing
up this point gives a variety that parameterizes genus-two curves with
$I_{2}\not=0$ and their degenerations. In the blow-up space we have to
introduce additional coordinates that are obtained as ratios of
$\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3}$ and have weight zero. Those are
precisely the coordinates $\mathbf{y}_{1},\mathbf{y}_{2},\mathbf{y}_{3}$
already introduced in Equation (11). It turns out that the coordinate ring of
the blown-up space is
$\mathbb{C}[\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3},\mathbf{y}_{1},\mathbf{y}_{2},\mathbf{y}_{3}]$.

If a Jacobian variety corresponds to a product of elliptic curves then
$\underline{\tau}$ is equivalent to a point with $z=0$, i.e.,
$\underline{\tau}$ is located on the Humbert surface $H_{1}$. We then have
$\chi_{10}(\underline{\tau})=0,\chi_{12}(\underline{\tau})\not=0$ and
$[I_{2}:I_{4}:I_{6}:I_{10}]=[1:0:0:0]$. Equations (12) and (13) imply
$\mathbf{x}_{1}=\mathbf{x}_{2}=\mathbf{x}_{3}=\mathbf{y}_{3}=0$ and
$\mathbf{y}_{1}=j(\tau_{1})\,j(\tau_{2})$ and
$\mathbf{y}_{2}=(1728-j(\tau_{1}))\,(1728-j(\tau_{2}))$.

##  4\. A universal genus-two curve from the moduli space

We start with a point $p\in{\mathcal{M}}_{2}$. This point can be given by
$\mathfrak{p}=[j_{1},j_{2},j_{3},1]$. The goal of this section is to
explicitly determine the equation of a genus-two curve over its minimal field
of definition corresponding to this point $\mathfrak{p}$.

Associated to the symmetric matrix $M$ whose determinant is given by $R^{2}$
in Equation (22) is the plane conic $\mathcal{Q}$ in the variables
$[X_{1}:X_{2}:X_{3}]\in\mathbb{P}^{2}$ given by

(30)
$A_{11}\,X_{1}^{2}+A_{22}\,X_{2}^{2}+A_{33}\,X_{3}^{2}+2A_{12}\,X_{1}X_{2}+2A_{13}\,X_{1}X_{3}+2A_{23}\,X_{2}X_{3}=0\;.$

Over the field extension $\mathbb{Q}[d]$ where $d$ is given by

(31) $d^{2}=-2\,A_{2,2}\,R^{2}\;$

we can re-express $A_{1,1}$ in terms of $d$ and the other coefficients as

(32)
$A_{1,1}=-\frac{d^{2}}{(A_{2,2}A_{3,3}-A_{2,3}^{2})\,A_{2,2}}+\frac{A_{1,2}^{2}A_{3,3}-2A_{1,2}A_{1,3}A_{2,3}+A_{1,3}^{2}A_{2,2}}{A_{2,2}A_{3,3}-A_{2,3}^{2}}\;.$

We can then find a rational point of the conic $\mathcal{Q}$ over
$\mathbb{Q}[d,A,B,C,D]$. In fact, a rational point
$[X^{(0)}_{1}:X^{(0)}_{2}:X^{(0)}_{3}]$ on the conic $\mathcal{Q}$ is given by

(33) $\begin{split}X^{(0)}_{1}&=A_{2,2}(A_{2,2}A_{3,3}-A_{2,3}^{2})\;,\\\
X^{(0)}_{2}&=\mp
A_{2,3}d-A_{2,2}\big{(}A_{1,2}A_{3,3}-A_{1,3}A_{2,3}\big{)}\;,\\\
X^{(0)}_{3}&=A_{2,2}\,\left(\pm
d+A_{1,2}A_{2,3}-A_{1,3}A_{2,2}\right)\;.\end{split}$

We substitute

(34)
$\begin{split}X^{(0)}_{3}X_{2}&=X^{(0)}_{2}X_{3}+t\,\big{(}X^{(0)}_{3}X_{1}-X^{(0)}_{1}X_{3}\big{)}\end{split}$

into Equation (30). We already know one of the roots of this quadratic, since
it must be satisfied if
$[X_{1}:X_{2}:X_{3}]=[X^{(0)}_{1}:X^{(0)}_{2}:X^{(0)}_{3}]$. The second root
is given by

(35)
$\begin{split}X_{1}=&A_{2,2}^{3}(A_{2,2}A_{3,3}-A_{2,3}^{2})^{2}\,t^{2}+2A_{1,2}A_{2,2}^{2}(A_{2,2}A_{3,3}-A_{2,3}^{2})\,t\\\
+&A_{2,2}(A_{2,2}A_{3,3}-A_{2,3}^{2})\left(A_{1,2}^{2}A_{2,2}A_{3,3}-2A_{1,2}A_{1,3}A_{2,2}A_{2,3}\right.\\\
&\left.+A_{1,3}^{2}A_{2,2}^{2}\pm
2(A_{1,2}A_{2,3}-A_{1,3}A_{2,2})\,d+d^{2}\right),\\\
X_{2}=&-A_{2,2}^{2}(A_{2,2}A_{3,3}-A_{2,3}^{2})(A_{1,2}A_{2,2}A_{3,3}-A_{1,3}A_{2,2}A_{2,3}\pm
A_{2,3}d)\,t^{2}\\\ +&2A_{2,2}(A_{2,2}A_{3,3}-A_{2,3}^{2})\\\
&\times(-A_{1,2}^{2}A_{2,2}A_{3,3}+A_{1,2}A_{1,3}A_{2,2}A_{2,3}\mp
A_{1,3}A_{2,2}d+d^{2})\,t\\\ +&(A_{1,2}A_{2,2}A_{3,3}-A_{1,3}A_{2,2}A_{2,3}\pm
A_{2,3}d)\\\
&\times(-A_{1,2}^{2}A_{2,2}A_{3,3}+2A_{1,2}A_{1,3}A_{2,2}A_{2,3}-A^{2}_{1,3}A^{2}_{2,2}+d^{2}),\\\
X_{3}=&A_{2,2}^{3}(A_{2,2}A_{3,3}-A_{2,3}^{2})(A_{1,2}A_{2,3}-A_{1,3}A_{2,2}\pm
d)\,t^{2}\\\
+&2A_{1,2}A_{2,2}^{2}(A_{2,2}A_{3,3}-A_{2,3}^{2})(A_{1,2}A_{2,3}-A_{1,3}A_{2,2}\pm
d)\,t\\\ -&A_{2,2}(A_{1,2}A_{2,3}-A_{1,3}A_{2,2}\pm d)\\\
&\times(-A_{1,2}^{2}A_{2,2}A_{3,3}+2A_{1,2}A_{1,3}A_{2,2}A_{2,3}-A^{2}_{1,3}A^{2}_{2,2}+d^{2}).\end{split}$

Using $A_{1,3}=A_{2,2}$ and $d^{2}=-2\,A_{2,2}\,R^{2}$ the point
$[X_{1}:X_{2}:X_{3}]$ is easily shown to be equivalent to

(36)
$\begin{split}&X_{1}=A_{2,2}^{2}(A_{2,2}A_{3,3}-A_{2,3}^{2})^{2}\,t^{2}+2A_{1,2}A_{2,2}(A_{2,2}A_{3,3}-A_{2,3}^{2})\,t\\\
&-A_{1,1}A_{2,2}^{2}A_{3,3}+A_{1,1}A_{2,2}A_{2,3}^{2}+2A_{1,2}^{2}A_{2,2}A_{3,3}-4A_{1,2}A_{2,2}^{2}A_{2,3}\\\
&\pm 2\,(A_{1,2}A_{2,3}-A_{2,2})\,d,\\\
&X_{2}=-A_{2,2}(A_{1,2}A_{2,2}A_{3,3}-A_{2,2}^{2}A_{2,3}\pm
A_{2,3}d)\,t^{2}\\\
&-2A_{2,2}(A_{1,1}A_{2,2}A_{3,3}-A_{1,1}A_{2,3}^{2}+A_{1,2}A_{2,2}A_{2,3}-A_{2,2}^{3}\pm
A_{2,2}d)\,t\\\ &-A_{1,1}(A_{1,2}A_{2,2}A_{3,3}-A_{2,2}^{2}A_{2,3}\pm
A_{2,3}d),\\\ &X_{3}=A_{2,2}^{2}(A_{1,2}A_{2,3}-A_{2,2}^{2}\pm
d)\,t^{2}+2A_{1,2}A_{2,2}(A_{1,2}A_{2,3}-A_{2,2}^{2}\pm d)\,t\\\
&+A_{1,1}A_{2,2}(A_{1,2}A_{2,3}-A_{2,2}\pm d).\end{split}$

Equations (36) give for any $t\in\mathbb{Q}$ a rational parametrization of the
conic $\mathcal{Q}$ over $\mathbb{Q}[d,A,B,C,D]$.

Similarly, associated to the coefficients $(a_{ijk})$ in Equation (24) is a
plane cubic curve $\mathcal{T}$ in the variables
$[X_{1}:X_{2}:X_{3}]\in\mathbb{P}^{3}$ given by

(37)
$\begin{split}&a_{111}\,X_{1}^{3}+a_{222}\,X_{2}^{3}+a_{333}\,X_{3}^{3}+6\,a_{123}\,X_{1}X_{2}X_{3}\\\
+\,&3\,a_{112}\,X_{1}^{2}X_{2}+3\,a_{113}\,X_{1}^{2}X_{3}+3\,a_{122}\,X_{1}X_{2}^{2}+3\,a_{223}\,X_{2}^{2}X_{3}\\\
+\,&3\,a_{133}\,X_{1}X_{3}^{2}+3\,a_{233}\,X_{2}X_{3}^{2}=0\;.\end{split}$

Plugging the rational parametrization of the conic $\mathcal{Q}$ from
Equations (36) into the cubic $\mathcal{T}$ in Equation (37), one obtains the
ramificattion locus of sextic curve. The ramification locus is equivalent to

(38)
$0=\sum_{i=0}^{6}\,\underbrace{\,18^{-\lfloor\frac{i+1}{2}\rfloor}\,\kappa_{i}\,\left(\delta_{i}\left(54D\right)^{\lfloor\frac{i+1}{2}\rfloor}\pm
54\cdot
3^{\lfloor\frac{(i-3)^{2}}{2}\rfloor-3\lfloor\frac{(i-3)^{2}}{5}\rfloor}\;\epsilon_{i}\,(54D)^{\lfloor\frac{i}{2}\rfloor}\,d\right)}_{=:\,a^{\pm}_{6-i}}\,t^{i}\;,$

where $\delta_{i},\epsilon_{i}$ are irreducible polynomials in
$\mathbb{Z}[A,B,C,D]$ and $\kappa_{i}=1,12,15B$, 360, 15, 12, 1 for
$i=0,\dots,6$ such that $a^{\pm}_{6-i}\in\mathbb{Q}[d,A,B,C,D]$.

$(A,B,C,D)$ are given as polynomial in terms of the invariants
$(I_{2},I_{4},I_{6},I_{10})$ in Equations (17). Thus, we can express all
coefficients of the sextic as polynomials in
$\mathbb{Q}[d,I_{2},I_{4},I_{6},I_{10}]$, and we have

(39)
$d^{2}=\frac{I_{30}}{2^{7}\,3^{9}\,5^{10}}\,\,(9I_{2}^{5}+700I_{2}^{3}I_{4}-3600I_{2}^{2}I_{6}-12400I_{2}I_{4}^{2}+48000I_{4}I_{6}+10800000I_{10}).$

Notice that $d^{2}$ has two significant factors: one is $I_{30}$ which
correspond exactly to the locus of the curves with extra involutions, and the
other one is the Clebsch invariant $D$. Next we have our main result:

######  Theorem 4.

For every point $\mathfrak{p}\in{\mathcal{M}}_{2}$ such that
$\mathfrak{p}\in{\mathcal{M}}_{2}(k)$, for some number field $K$, there is a
pair of genus-two curves $\mathcal{C}^{\pm}$ given by

$\mathcal{C}^{\pm}:\quad y^{2}=\sum_{i=0}^{6}\,a_{6-i}^{\pm}\,x^{i}\;,$

corresponding to $\mathfrak{p}$, such that $a^{\pm}_{i}\in K(d)$,
$i=0,\dots,6$ as given explicitly in Equation (42). Moreover, $K(d)$ is the
minimal field of definition of $\mathfrak{p}$.

###### Proof.

From the above discussion we know that there is a genus-two curve
$\mathcal{C}:y^{2}=f(t)$ corresponding to $\mathfrak{p}$, where $f(t)$ is
given in Equation (38). The coefficients
$a_{i}^{\pm}\in\mathbb{Q}[d,I_{2},I_{4},I_{6},I_{10}]$ as displayed in
Equation (42). The field of moduli $K$ of the point $\mathfrak{p}$ is
$K=\mathbb{Q}(j_{1},j_{2},j_{3})$. Therefore, $a_{i}^{\pm}\in K\,[d]$, for
$i=0,\dots,6$. This completes the proof.

∎

Computing expressions for $a^{\pm}_{i}\in
K[d]=\mathbb{Q}(j_{1},j_{2},j_{3})[d]$ is straightforward, but we choose not
to display them here.

######  Corollary 5.

Let $j_{1},j_{2},j_{3}$ be transcendentals. There exists a pair of genus-two
curves $\mathcal{C}^{\pm}$ defined over $\mathbb{Q}(j_{1},j_{2},j_{3})[d]$
such that

$j_{1}(\mathcal{C}^{\pm})=j_{1},\quad j_{2}(\mathcal{C}^{\pm})=j_{2},\quad
j_{3}(\mathcal{C}^{\pm})=j_{3},$

where $d^{2}$ is given in terms of $(j_{1},j_{2},j_{3})$ in Equation (41).

The result in Theorem 4 answers completely the question; for what curves the
field of moduli is a field of definition. The answer is exactly for curves
such that $d$ is a complete square in the field of moduli $K$. Obviously, when
the curve has extra involutions, then $I_{30}=0$ and therefore the field of
moduli is a field of definition.

######  Corollary 6.

Let $D$ be the Clebsch invariant and $\mathfrak{p}\in{\mathcal{M}}_{2}$. If
$D(\mathfrak{p})=0$ then there is a genus-two curve $\mathcal{C}$ defined over
the field of moduli.

######  Remark 7.

The locus $D=0$ is given in terms of Siegel modular forms by

$\begin{split}0=&\,250\,\chi_{10}^{5}\psi_{4}\psi_{6}+675\,\chi_{10}^{4}\chi_{{12}}\psi_{4}^{2}+86400000\,\chi_{10}^{6}-2700\,\chi_{10}^{3}\chi_{12}^{2}\psi_{6}\\\
&-13500\,\chi_{10}^{2}\chi_{12}^{3}\psi_{4}+34992\,\chi_{12}^{5}\,,\end{split}$

and in terms of the absolute invariants $(j_{1},j_{2},j_{3})$ by

$\begin{split}0=&\,9\,j_{1}^{2}+700\,j_{2}j_{1}-3600\,j_{3}j_{1}-12400\,j_{2}^{2}+48000\,j_{2}j_{3}+10800000\,j_{1}\;.\end{split}$

Therefore, the locus $D=0$ is parametrized by $j_{1},j_{2}\in\mathbb{C}$ and
$j_{3}=h(j_{1},j_{2})$ with

$h(j_{1},j_{2})=3000+\frac{j_{1}}{400}+\frac{41\,j_{2}}{180}-\frac{j_{2}}{9}\cdot\frac{11\,j_{2}-1080000}{3\,j_{1}-40j_{2}}\;.$

Notice that $d=0$ is equivalent to $D\cdot\chi_{35}=0$. Hence, we have the
following observation:

######  Corollary 8.

Let $\mathfrak{p}\in{\mathcal{M}}_{2}$. If

$\begin{split}\chi_{35}\,\left(\,250\,\chi_{10}^{5}\psi_{4}\psi_{6}+675\,\chi_{10}^{4}\chi_{{12}}\psi_{4}^{2}+86400000\,\chi_{10}^{6}-2700\,\chi_{10}^{3}\chi_{12}^{2}\psi_{6}\right.\\\
\left.-13500\,\chi_{10}^{2}\chi_{12}^{3}\psi_{4}+34992\,\chi_{12}^{5}\right)&=0,\end{split}$

then the field of moduli of $\mathfrak{p}$ is also a field of definition.

The above theorem gives two twists of the curve $\mathcal{C}$ which have
moduli point $\mathfrak{p}$. Whether such equation can be modified so that it
can be minimal is unknown.

##  5\. Appendix

In Equations (42) we will display the polynomials $\delta_{i},\epsilon_{i}$
for $i=0,\dots,6$ that determine the pair of genus-two curves
$\mathcal{C}^{\pm}$ given by

(40)
$y^{2}=\sum_{i=0}^{6}\,a_{6-i}^{\pm}\,x^{i}=\sum_{i=0}^{6}\,18^{-\lfloor\frac{i+1}{2}\rfloor}\,\kappa_{i}\,\Big{(}\delta_{i}\left(54D\right)^{\lfloor\frac{i+1}{2}\rfloor}\pm
54\cdot
3^{\lfloor\frac{(i-3)^{2}}{2}\rfloor-3\lfloor\frac{(i-3)^{2}}{5}\rfloor}\;\epsilon_{i}\,(54D)^{\lfloor\frac{i}{2}\rfloor}\,d\Big{)}\,x^{i}$

where $(A,B,C,D)$ are the Clebsch invariants and
$\kappa_{i}=1,12,15B,360,15,12,1$ for $i=0,\dots,6$. The Clebsch invariants
are given as polynomial in terms of $(I_{2},I_{4},I_{6},I_{10})$ in Equations
(17). The square $d^{2}$ is given in terms of $(j_{1},j_{2},j_{3})$ by

$\displaystyle d^{2}$
$\displaystyle=\frac{1}{2^{22}3^{36}5^{30}\,j_{1}^{9}}\Big{(}j_{2}^{4}j_{1}^{3}-12\,j_{2}^{3}j_{3}j_{1}^{3}+54\,j_{2}^{2}j_{3}^{2}j_{1}^{3}-108\,j_{2}j_{3}^{3}j_{1}^{3}+81\,j_{3}^{4}j_{1}^{3}+78\,j_{2}^{5}j_{1}^{2}-1332\,j_{2}^{4}j_{3}j_{1}^{2}+8910\,j_{2}^{3}j_{3}^{2}j_{1}^{2}-29376\,j_{2}^{2}j_{3}^{3}j_{1}^{2}+47952\,j_{2}j_{3}^{4}j_{1}^{2}-31104\,j_{3}^{5}j_{1}^{2}$
(41)
$\displaystyle-159\,j_{2}^{6}j_{1}+1728\,j_{2}^{5}j_{3}j_{1}-6048\,j_{2}^{4}j_{3}^{2}j_{1}+6912\,j_{2}^{3}j_{3}^{3}j_{1}+80\,j_{2}^{7}-384\,j_{2}^{6}j_{3}-972\,j_{2}^{2}j_{1}^{4}+5832\,j_{2}j_{3}j_{1}^{4}-8748\,j_{3}^{2}j_{1}^{4}-77436\,j_{2}^{3}j_{1}^{3}+870912\,j_{2}^{2}j_{3}j_{1}^{3}$
$\displaystyle-3090960\,j_{2}j_{3}^{2}j_{1}^{3}+3499200\,j_{3}^{3}j_{1}^{3}+592272\,j_{2}^{4}j_{1}^{2}-4743360\,j_{2}^{3}j_{3}j_{1}^{2}+9331200\,j_{2}^{2}j_{3}^{2}j_{1}^{2}-41472\,j_{2}^{5}j_{1}+236196\,j_{1}^{5}+19245600\,j_{2}j_{1}^{4}-104976000\,j_{3}j_{1}^{4}$
$\displaystyle-507384000\,j_{2}^{2}j_{1}^{3}+2099520000\,j_{2}j_{3}j_{1}^{3}+125971200000\,j_{1}^{4}\Big{)}\Big{(}9\,j_{1}^{2}+700\,j_{2}j_{1}-3600\,j_{3}j_{1}-12400\,j_{2}^{2}+48000\,j_{2}j_{3}+10800000\,j_{1}\Big{)}.$

The irreducible polynomials $\delta_{i},\epsilon_{i}$ in $\mathbb{Z}[A,B,C,D]$
for $i=0,\dots,6$ are given by

$\displaystyle\delta_{6}$
$\displaystyle=-2048\,{A}^{5}{B}^{3}{C}^{5}-9216\,{A}^{4}{B}^{5}{C}^{4}-16128\,{A}^{3}{B}^{7}{C}^{3}-13824\,{A}^{2}{B}^{9}{C}^{2}-5832\,A{B}^{11}C-972\,{B}^{13}-9216\,{A}^{4}{B}^{2}{C}^{6}-36096\,{A}^{3}{B}^{4}{C}^{5}$
$\displaystyle-50112\,{A}^{2}{B}^{6}{C}^{4}-29808\,A{B}^{8}{C}^{3}-6480\,{B}^{10}{C}^{2}-6912\,{A}^{4}B{C}^{5}D-35712\,{A}^{3}{B}^{3}{C}^{4}D-13824\,{C}^{7}{A}^{3}B-55728\,{A}^{2}{B}^{5}{C}^{3}D-48384\,{C}^{6}{A}^{2}{B}^{3}$
$\displaystyle-34992\,A{B}^{7}{C}^{2}D-49248\,{C}^{5}A{B}^{5}-7776\,{B}^{9}CD-15552\,{C}^{4}{B}^{7}+25920\,{A}^{3}{B}^{2}{C}^{3}{D}^{2}-10368\,{A}^{3}{C}^{6}D+54432\,{A}^{2}{B}^{4}{C}^{2}{D}^{2}-90720\,{A}^{2}{B}^{2}{C}^{5}D$
$\displaystyle-6912\,{C}^{8}{A}^{2}+37908\,A{B}^{6}C{D}^{2}-114048\,A{B}^{4}{C}^{4}D-25920\,{C}^{7}A{B}^{2}+8748\,{B}^{8}{D}^{2}-38880\,{B}^{6}{C}^{3}D-15552\,{C}^{6}{B}^{4}+136080\,{A}^{2}B{C}^{4}{D}^{2}$
$\displaystyle+208008\,A{B}^{3}{C}^{3}{D}^{2}-108864\,AB{C}^{6}D+79704\,{B}^{5}{C}^{2}{D}^{2}-77760\,{B}^{3}{C}^{5}D-5184\,{C}^{8}B+7776\,{A}^{2}{C}^{3}{D}^{3}+46656\,A{B}^{2}{C}^{2}{D}^{3}+139968\,A{C}^{5}{D}^{2}$
(42)
$\displaystyle+34992\,{B}^{4}C{D}^{3}+116640\,{B}^{2}{C}^{4}{D}^{2}-62208\,{C}^{7}D-52488\,ABC{D}^{4}-19683\,{B}^{3}{D}^{4}+23328\,B{C}^{3}{D}^{3}-139968\,{C}^{2}{D}^{4},$
$\displaystyle\epsilon_{6}$
$\displaystyle=-128\,{A}^{3}{B}^{2}{C}^{3}-288\,{A}^{2}{B}^{4}{C}^{2}-216\,A{B}^{6}C-54\,{B}^{8}-384\,{C}^{4}{A}^{2}B-576\,{C}^{3}A{B}^{3}-216\,{B}^{5}{C}^{2}-48\,{A}^{2}{C}^{3}D+108\,A{B}^{2}{C}^{2}D$
$\displaystyle-288\,{C}^{5}A+108\,{B}^{4}CD-216\,{B}^{2}{C}^{4}+324\,ABC{D}^{2}+243\,{B}^{3}{D}^{2}+288\,B{C}^{3}D+864\,{C}^{2}{D}^{2},$
$\displaystyle\delta_{5}$
$\displaystyle=1024\,{A}^{5}{B}^{4}{C}^{4}+3072\,{A}^{4}{B}^{6}{C}^{3}+3456\,{A}^{3}{B}^{8}{C}^{2}+1728\,{A}^{2}{B}^{10}C+324\,A{B}^{12}+8064\,{A}^{4}{B}^{3}{C}^{5}+20352\,{A}^{3}{B}^{5}{C}^{4}+18648\,{A}^{2}{B}^{7}{C}^{3}$
$\displaystyle+7236\,A{B}^{9}{C}^{2}+972\,C{B}^{11}-6912\,{A}^{4}{B}^{2}{C}^{4}D-25056\,{A}^{3}{B}^{4}{C}^{3}D+20736\,{C}^{6}{A}^{3}{B}^{2}-32832\,{A}^{2}{B}^{6}{C}^{2}D+38880\,{C}^{5}{A}^{2}{B}^{4}-18630\,A{B}^{8}CD$
$\displaystyle+23544\,{C}^{4}A{B}^{6}-3888\,{B}^{10}D+4536\,{C}^{3}{B}^{8}-5184\,{A}^{3}{B}^{3}{C}^{2}{D}^{2}-17712\,{A}^{3}B{C}^{5}D-7776\,{A}^{2}{B}^{5}C{D}^{2}-56376\,{A}^{2}{B}^{3}{C}^{4}D+21600\,{C}^{7}{A}^{2}B$
$\displaystyle-2916\,A{B}^{7}{D}^{2}-53784\,A{B}^{5}{C}^{3}D+24624\,{C}^{6}A{B}^{3}-16092\,{B}^{7}{C}^{2}D+6480\,{C}^{5}{B}^{5}-3888\,{A}^{3}{C}^{4}{D}^{2}-46656\,{A}^{2}{B}^{2}{C}^{3}{D}^{2}-11664\,{A}^{2}{C}^{6}D$
$\displaystyle-51030\,A{B}^{4}{C}^{2}{D}^{2}-40824\,A{B}^{2}{C}^{5}D+7776\,{C}^{8}A-13608\,{B}^{6}C{D}^{2}-23328\,{B}^{4}{C}^{4}D+2592\,{C}^{7}{B}^{2}+48600\,{A}^{2}B{C}^{2}{D}^{3}+83835\,A{B}^{3}C{D}^{3}$
$\displaystyle-78732\,AB{C}^{4}{D}^{2}+34992\,{B}^{5}{D}^{3}-49572\,{B}^{3}{C}^{3}{D}^{2}-11664\,B{C}^{6}D+6561\,A{B}^{2}{D}^{4}+96228\,A{C}^{3}{D}^{3}+97686\,{B}^{2}{C}^{2}{D}^{3}-52488\,{C}^{5}{D}^{2}$
$\displaystyle+41553\,BC{D}^{4}-78732\,{D}^{5},$ $\displaystyle\epsilon_{5}$
$\displaystyle=128\,{A}^{3}{B}^{4}{C}^{3}+288\,{A}^{2}{B}^{6}{C}^{2}+216\,A{B}^{8}C+54\,{B}^{10}+576\,{A}^{2}{B}^{3}{C}^{4}+864\,A{B}^{5}{C}^{3}+324\,{B}^{7}{C}^{2}+1296\,{A}^{2}{B}^{2}{C}^{3}D+2052\,A{B}^{4}{C}^{2}D$
$\displaystyle+864\,{C}^{5}A{B}^{2}+810\,{B}^{6}CD+648\,{C}^{4}{B}^{4}-3456\,{A}^{2}B{C}^{2}{D}^{2}-5508\,A{B}^{3}C{D}^{2}+3240\,AB{C}^{4}D-2187\,{B}^{5}{D}^{2}+2592\,{B}^{3}{C}^{3}D+432\,{C}^{6}B$
$\displaystyle-4860\,A{C}^{3}{D}^{2}-4050\,{B}^{2}{C}^{2}{D}^{2}+1944\,{C}^{5}D-243\,BC{D}^{3}+8748\,{D}^{4},$
$\displaystyle\delta_{4}$
$\displaystyle=2048\,{A}^{6}{B}^{4}{C}^{5}+7168\,{A}^{5}{B}^{6}{C}^{4}+9984\,{A}^{4}{B}^{8}{C}^{3}+6912\,{A}^{3}{B}^{10}{C}^{2}+2376\,{A}^{2}{B}^{12}C+324\,A{B}^{14}+15360\,{A}^{5}{B}^{3}{C}^{6}+40704\,{A}^{4}{B}^{5}{C}^{5}$
$\displaystyle+38208\,{A}^{3}{B}^{7}{C}^{4}+13392\,{A}^{2}{B}^{9}{C}^{3}-648\,{B}^{13}C-20736\,{A}^{5}{B}^{2}{C}^{5}D-93312\,{A}^{4}{B}^{4}{C}^{4}D+41472\,{A}^{4}{C}^{7}{B}^{2}-167184\,{A}^{3}{B}^{6}{C}^{3}D$
$\displaystyle+66816\,{A}^{3}{C}^{6}{B}^{4}-149040\,{A}^{2}{B}^{8}{C}^{2}D+16416\,{A}^{2}{C}^{5}{B}^{6}-66096\,A{B}^{10}CD-18144\,A{B}^{8}{C}^{4}-11664\,{B}^{12}D-7776\,{B}^{10}{C}^{3}-5184\,{A}^{4}{B}^{3}{C}^{3}{D}^{2}$
$\displaystyle-51840\,{A}^{4}B{C}^{6}D-12960\,{A}^{3}{B}^{5}{C}^{2}{D}^{2}-194400\,{A}^{3}{B}^{3}{C}^{5}D+48384\,{A}^{3}B{C}^{8}-10692\,{A}^{2}{B}^{7}C{D}^{2}-272160\,{A}^{2}{B}^{5}{C}^{4}D+8640\,{A}^{2}{B}^{3}{C}^{7}$
$\displaystyle-2916\,A{B}^{9}{D}^{2}-168480\,A{B}^{7}{C}^{3}D-62208\,A{B}^{5}{C}^{6}-38880\,{B}^{9}{C}^{2}D-31104\,{B}^{7}{C}^{5}-10368\,{A}^{4}{C}^{5}{D}^{2}-81648\,{A}^{3}{B}^{2}{C}^{4}{D}^{2}-31104\,{A}^{3}{C}^{7}D$
$\displaystyle-141912\,{A}^{2}{B}^{4}{C}^{3}{D}^{2}-93312\,{A}^{2}{B}^{2}{C}^{6}D+20736\,{C}^{9}{A}^{2}-88128\,A{B}^{6}{C}^{2}{D}^{2}-93312\,A{B}^{4}{C}^{5}D-57024\,{C}^{8}A{B}^{2}-17496\,{B}^{8}C{D}^{2}-31104\,{B}^{6}{C}^{4}D$
$\displaystyle-51840\,{C}^{7}{B}^{4}+132192\,{A}^{3}B{C}^{3}{D}^{3}+369360\,{A}^{2}{B}^{3}{C}^{2}{D}^{3}-139968\,{A}^{2}B{C}^{5}{D}^{2}+344088\,A{B}^{5}C{D}^{3}-221616\,A{B}^{3}{C}^{4}{D}^{2}+104976\,{B}^{7}{D}^{3}$
$\displaystyle-81648\,{B}^{5}{C}^{3}{D}^{2}-31104\,{C}^{9}B+256608\,{A}^{2}{C}^{4}{D}^{3}+6561\,A{B}^{4}{D}^{4}+501552\,A{B}^{2}{C}^{3}{D}^{3}-139968\,A{C}^{6}{D}^{2}+268272\,{B}^{4}{C}^{2}{D}^{3}-139968\,{B}^{2}{C}^{5}{D}^{2}$
$\displaystyle+52488\,AB{C}^{2}{D}^{4}+91854\,{B}^{3}C{D}^{4}-69984\,B{C}^{4}{D}^{3}-209952\,AC{D}^{5}-236196\,{B}^{2}{D}^{5},$
$\displaystyle\epsilon_{4}$
$\displaystyle=-128\,{A}^{4}{B}^{3}{C}^{3}-288\,{A}^{3}{B}^{5}{C}^{2}-216\,{A}^{2}{B}^{7}C-54\,A{B}^{9}+576\,{A}^{2}{B}^{4}{C}^{3}+864\,A{B}^{6}{C}^{2}+324\,{B}^{8}C-1584\,{A}^{3}B{C}^{3}D-3924\,{A}^{2}{B}^{3}{C}^{2}D$
$\displaystyle+864\,{A}^{2}B{C}^{5}-3348\,A{B}^{5}CD+2376\,A{B}^{3}{C}^{4}-972\,{B}^{7}D+1296\,{B}^{5}{C}^{3}+324\,{A}^{2}{B}^{2}C{D}^{2}-2160\,{A}^{2}{C}^{4}D+243\,A{B}^{4}{D}^{2}-2808\,A{B}^{2}{C}^{3}D$
$\displaystyle+864\,{C}^{6}A-1080\,{B}^{4}{C}^{2}D+1296\,{C}^{5}{B}^{2}+972\,AB{C}^{2}{D}^{2}+486\,{B}^{3}C{D}^{2}+1296\,B{C}^{4}D+3888\,AC{D}^{3}+4374\,{B}^{2}{D}^{3},$
$\displaystyle\delta_{3}$
$\displaystyle=-512\,{A}^{6}{B}^{2}{C}^{6}-3456\,{A}^{5}{B}^{4}{C}^{5}-8864\,{A}^{4}{B}^{6}{C}^{4}-11464\,{A}^{3}{B}^{8}{C}^{3}-8028\,{A}^{2}{B}^{10}{C}^{2}-2916\,A{B}^{12}C-432\,{B}^{14}+384\,{A}^{5}{B}^{3}{C}^{4}D$
$\displaystyle-1536\,{A}^{5}B{C}^{7}+864\,{A}^{4}{B}^{5}{C}^{3}D-11904\,{A}^{4}{B}^{3}{C}^{6}+648\,{A}^{3}{B}^{7}{C}^{2}D-28320\,{A}^{3}{B}^{5}{C}^{5}+162\,{A}^{2}{B}^{9}CD-29976\,{A}^{2}{B}^{7}{C}^{4}-14832\,A{B}^{9}{C}^{3}$
$\displaystyle-2808\,{B}^{11}{C}^{2}-192\,{A}^{5}{C}^{6}D+3024\,{A}^{4}{B}^{2}{C}^{5}D-1152\,{A}^{4}{C}^{8}+6120\,{A}^{3}{B}^{4}{C}^{4}D-16416\,{A}^{3}{B}^{2}{C}^{7}+3768\,{A}^{2}{B}^{6}{C}^{3}D-35856\,{A}^{2}{B}^{4}{C}^{6}$
$\displaystyle+720\,A{B}^{8}{C}^{2}D-27936\,A{B}^{6}{C}^{5}-7344\,{B}^{8}{C}^{4}+2592\,{A}^{4}B{C}^{4}{D}^{2}+15120\,{A}^{3}{B}^{3}{C}^{3}{D}^{2}+10224\,{A}^{3}B{C}^{6}D+25434\,{A}^{2}{B}^{5}{C}^{2}{D}^{2}+12312\,{A}^{2}{B}^{3}{C}^{5}D$
$\displaystyle-12960\,{A}^{2}B{C}^{8}+16848\,A{B}^{7}C{D}^{2}+3024\,A{B}^{5}{C}^{4}D-22464\,A{B}^{3}{C}^{7}+3888\,{B}^{9}{D}^{2}-360\,{B}^{7}{C}^{3}D-9504\,{B}^{5}{C}^{6}-972\,{A}^{3}{B}^{2}{C}^{2}{D}^{3}$
$\displaystyle+6048\,{A}^{3}{C}^{5}{D}^{2}-729\,{A}^{2}{B}^{4}C{D}^{3}+44388\,{A}^{2}{B}^{2}{C}^{4}{D}^{2}+7776\,{A}^{2}{C}^{7}D+58968\,A{B}^{4}{C}^{3}{D}^{2}-2592\,A{B}^{2}{C}^{6}D-5184\,A{C}^{9}+21600\,{B}^{6}{C}^{2}{D}^{2}$
$\displaystyle-5184\,{B}^{4}{C}^{5}D-5184\,{B}^{2}{C}^{8}-14580\,{A}^{2}B{C}^{3}{D}^{3}-15552\,A{B}^{3}{C}^{2}{D}^{3}+34992\,AB{C}^{5}{D}^{2}-3888\,{B}^{5}C{D}^{3}+29160\,{B}^{3}{C}^{4}{D}^{2}-7776\,B{C}^{7}D$
$\displaystyle-3888\,{A}^{2}{C}^{2}{D}^{4}-16767\,A{B}^{2}C{D}^{4}-29160\,A{C}^{4}{D}^{3}-8748\,{B}^{4}{D}^{4}-20412\,{B}^{2}{C}^{3}{D}^{3}+11664\,{C}^{6}{D}^{2}-24786\,B{C}^{2}{D}^{4}+17496\,C{D}^{5},$
$\displaystyle\epsilon_{3}$
$\displaystyle=-512\,{A}^{5}{B}^{3}{C}^{4}-1536\,{A}^{4}{B}^{5}{C}^{3}-1728\,{A}^{3}{B}^{7}{C}^{2}-864\,{A}^{2}{B}^{9}C-162\,A{B}^{11}-2304\,{A}^{4}{C}^{5}{B}^{2}-4416\,{A}^{3}{C}^{4}{B}^{4}-2160\,{A}^{2}{C}^{3}{B}^{6}+324\,A{C}^{2}{B}^{8}$
$\displaystyle+324\,C{B}^{10}-1728\,{A}^{4}B{C}^{4}D-8064\,{A}^{3}{B}^{3}{C}^{3}D-3456\,{A}^{3}B{C}^{6}-12636\,{A}^{2}{B}^{5}{C}^{2}D-1728\,{A}^{2}{B}^{3}{C}^{5}-8262\,A{B}^{7}CD+3240\,A{B}^{5}{C}^{4}-1944\,{B}^{9}D$
$\displaystyle+1944\,{B}^{7}{C}^{3}+1296\,{A}^{3}{B}^{2}{C}^{2}{D}^{2}-2592\,{A}^{3}{C}^{5}D+1944\,{A}^{2}{B}^{4}C{D}^{2}-14904\,{A}^{2}{B}^{2}{C}^{4}D-1728\,{C}^{7}{A}^{2}+729\,A{B}^{6}{D}^{2}-18792\,A{B}^{4}{C}^{3}D$
$\displaystyle+3888\,{C}^{6}A{B}^{2}-6804\,{B}^{6}{C}^{2}D+3888\,{C}^{5}{B}^{4}+12636\,{A}^{2}B{C}^{3}{D}^{2}+17982\,A{B}^{3}{C}^{2}{D}^{2}-9720\,AB{C}^{5}D+6318\,{B}^{5}C{D}^{2}-7776\,{B}^{3}{C}^{4}D+2592\,B{C}^{7}$
$\displaystyle+3888\,{A}^{2}{C}^{2}{D}^{3}+15309\,A{B}^{2}C{D}^{3}+17496\,A{C}^{4}{D}^{2}+8748\,{B}^{4}{D}^{3}+14580\,{B}^{2}{C}^{3}{D}^{2}-3888\,{C}^{6}D+16038\,B{C}^{2}{D}^{3}-17496\,C{D}^{4},$
$\displaystyle\delta_{2}$
$\displaystyle=6144\,{A}^{7}{B}^{4}{C}^{5}+23552\,{A}^{6}{B}^{6}{C}^{4}+36096\,{A}^{5}{B}^{8}{C}^{3}+27648\,{A}^{4}{B}^{10}{C}^{2}+10584\,{A}^{3}{B}^{12}C+1620\,{A}^{2}{B}^{14}+39936\,{A}^{6}{C}^{6}{B}^{3}+80640\,{A}^{5}{B}^{5}{C}^{5}$
$\displaystyle-17856\,{A}^{4}{B}^{7}{C}^{4}-168432\,{A}^{3}{B}^{9}{C}^{3}-168048\,{A}^{2}{B}^{11}{C}^{2}-68688\,A{B}^{13}C-10368\,{B}^{15}-34560\,{A}^{6}{B}^{2}{C}^{5}D-114048\,{A}^{5}{B}^{4}{C}^{4}D+96768\,{A}^{5}{C}^{7}{B}^{2}$
$\displaystyle-140400\,{A}^{4}{B}^{6}{C}^{3}D-76032\,{A}^{4}{B}^{4}{C}^{6}-76464\,{A}^{3}{B}^{8}{C}^{2}D-733536\,{A}^{3}{B}^{6}{C}^{5}-15552\,{A}^{2}{B}^{10}CD-981504\,{A}^{2}{B}^{8}{C}^{4}-515808\,A{B}^{10}{C}^{3}$
$\displaystyle-97200\,{B}^{12}{C}^{2}-36288\,{A}^{5}{B}^{3}{C}^{3}{D}^{2}-93312\,{A}^{5}B{C}^{6}D-80352\,{A}^{4}{B}^{5}{C}^{2}{D}^{2}+75168\,{A}^{4}{B}^{3}{C}^{5}D+103680\,{A}^{4}{C}^{8}B-59292\,{A}^{3}{B}^{7}C{D}^{2}$
$\displaystyle+803520\,{A}^{3}{B}^{5}{C}^{4}D-675648\,{A}^{3}{B}^{3}{C}^{7}-14580\,{A}^{2}{B}^{9}{D}^{2}+1143072\,{A}^{2}{B}^{7}{C}^{3}D-1829952\,{A}^{2}{B}^{5}{C}^{6}+632448\,A{B}^{9}{C}^{2}D-1425600\,A{B}^{7}{C}^{5}$
$\displaystyle+124416\,{B}^{11}CD-357696\,{B}^{9}{C}^{4}-20736\,{A}^{5}{C}^{5}{D}^{2}-423792\,{A}^{4}{B}^{2}{C}^{4}{D}^{2}-62208\,{A}^{4}{C}^{7}D-740664\,{A}^{3}{B}^{4}{C}^{3}{D}^{2}+1041984\,{A}^{3}{B}^{2}{C}^{6}D$
$\displaystyle+41472\,{A}^{3}{C}^{9}-434808\,{A}^{2}{B}^{6}{C}^{2}{D}^{2}+2799360\,{A}^{2}{B}^{4}{C}^{5}D-1104192\,{A}^{2}{B}^{2}{C}^{8}-81648\,A{B}^{8}C{D}^{2}+2317248\,A{B}^{6}{C}^{4}D-1710720\,A{B}^{4}{C}^{7}$
$\displaystyle+622080\,{B}^{8}{C}^{3}D-642816\,{B}^{6}{C}^{6}+256608\,{A}^{4}B{C}^{3}{D}^{3}+412128\,{A}^{3}{B}^{3}{C}^{2}{D}^{3}-855360\,{A}^{3}B{C}^{5}{D}^{2}+163296\,{A}^{2}{B}^{5}C{D}^{3}-746496\,{A}^{2}{B}^{3}{C}^{4}{D}^{2}$
$\displaystyle+1399680\,{A}^{2}B{C}^{7}D+128304\,A{B}^{5}{C}^{3}{D}^{2}+2426112\,A{B}^{3}{C}^{6}D-746496\,AB{C}^{9}+159408\,{B}^{7}{C}^{2}{D}^{2}+1026432\,{B}^{5}{C}^{5}D-559872\,{B}^{3}{C}^{8}$
$\displaystyle+52488\,{A}^{3}{B}^{2}C{D}^{4}+419904\,{A}^{3}{C}^{4}{D}^{3}+32805\,{A}^{2}{B}^{4}{D}^{4}-1539648\,{A}^{2}{B}^{2}{C}^{3}{D}^{3}-699840\,{A}^{2}{C}^{6}{D}^{2}-2939328\,A{B}^{4}{C}^{2}{D}^{3}+489888\,A{B}^{2}{C}^{5}{D}^{2}$
$\displaystyle+559872\,A{C}^{8}D-1119744\,{B}^{6}C{D}^{3}+629856\,{B}^{4}{C}^{4}{D}^{2}+559872\,{B}^{2}{C}^{7}D-186624\,{C}^{10}+1364688\,{A}^{2}B{C}^{2}{D}^{4}+1758348\,A{B}^{3}C{D}^{4}$
$\displaystyle-4199040\,AB{C}^{4}{D}^{3}+629856\,{B}^{5}{D}^{4}-3359232\,{B}^{3}{C}^{3}{D}^{3}+839808\,B{C}^{6}{D}^{2}-419904\,{A}^{2}C{D}^{5}+2519424\,A{C}^{3}{D}^{4}+1810836\,{B}^{2}{C}^{2}{D}^{4}$
$\displaystyle-1679616\,{C}^{5}{D}^{3}+2519424\,BC{D}^{5}-1889568\,{D}^{6},$
$\displaystyle\epsilon_{2}$
$\displaystyle=128\,{A}^{5}{B}^{3}{C}^{3}+288\,{A}^{4}{B}^{5}{C}^{2}+216\,{A}^{3}{B}^{7}C+54\,{A}^{2}{B}^{9}+1152\,{A}^{4}{C}^{4}{B}^{2}+2112\,{A}^{3}{C}^{3}{B}^{4}+1224\,{A}^{2}{C}^{2}{B}^{6}+216\,AC{B}^{8}-3024\,{A}^{4}B{C}^{3}D$
$\displaystyle-5868\,{A}^{3}{B}^{3}{C}^{2}D+2592\,{A}^{3}{C}^{5}B-3564\,{A}^{2}{B}^{5}CD+1944\,{A}^{2}{C}^{4}{B}^{3}-648\,A{B}^{7}D-864\,A{C}^{3}{B}^{5}-648\,{B}^{7}{C}^{2}-324\,{A}^{3}{B}^{2}C{D}^{2}-4320\,{A}^{3}{C}^{4}D$
$\displaystyle-243\,{A}^{2}{B}^{4}{D}^{2}+11664\,{A}^{2}{B}^{2}{C}^{3}D+1728\,{A}^{2}{C}^{6}+24624\,A{B}^{4}{C}^{2}D-2592\,A{C}^{5}{B}^{2}+9936\,{B}^{6}CD-2592\,{B}^{4}{C}^{4}-13608\,{A}^{2}B{C}^{2}{D}^{2}$
$\displaystyle-20412\,A{B}^{3}C{D}^{2}+34992\,AB{C}^{4}D-7776\,{B}^{5}{D}^{2}+27216\,{B}^{3}{C}^{3}D-2592\,B{C}^{6}+7776\,{A}^{2}C{D}^{3}+2916\,A{B}^{2}{D}^{3}-23328\,A{C}^{3}{D}^{2}-20412\,{B}^{2}{C}^{2}{D}^{2}$
$\displaystyle+15552\,{C}^{5}D-29160\,BC{D}^{3}+34992\,{D}^{4},$
$\displaystyle\delta_{1}$
$\displaystyle=-1024\,{A}^{7}{B}^{6}{C}^{4}-3072\,{A}^{6}{B}^{8}{C}^{3}-3456\,{A}^{5}{B}^{10}{C}^{2}-1728\,{A}^{4}{B}^{12}C-324\,{A}^{3}{B}^{14}+3072\,{A}^{7}{C}^{6}{B}^{3}+31104\,{A}^{6}{B}^{5}{C}^{5}+120384\,{A}^{5}{B}^{7}{C}^{4}$
$\displaystyle+209496\,{A}^{4}{B}^{9}{C}^{3}+181764\,{A}^{3}{B}^{11}{C}^{2}+77436\,{A}^{2}{B}^{13}C+12960\,{B}^{15}A+41472\,{A}^{6}{B}^{4}{C}^{4}D+82944\,{A}^{6}{C}^{7}{B}^{2}+130464\,{A}^{5}{B}^{6}{C}^{3}D$
$\displaystyle+705024\,{A}^{5}{C}^{6}{B}^{4}+153360\,{A}^{4}{B}^{8}{C}^{2}D+2112480\,{A}^{4}{C}^{5}{B}^{6}+79866\,{A}^{3}{B}^{10}CD+2975832\,{A}^{3}{B}^{8}{C}^{4}+15552\,{A}^{2}{B}^{12}D+2151144\,{A}^{2}{B}^{10}{C}^{3}$
$\displaystyle+772416\,A{B}^{12}{C}^{2}+108864\,{B}^{14}C+38016\,{A}^{6}B{C}^{6}D+5184\,{A}^{5}{B}^{5}{C}^{2}{D}^{2}+352080\,{A}^{5}{B}^{3}{C}^{5}D+228096\,{A}^{5}{C}^{8}B+7776\,{A}^{4}{B}^{7}C{D}^{2}$
$\displaystyle+456840\,{A}^{4}{B}^{5}{C}^{4}D+2178144\,{A}^{4}{C}^{7}{B}^{3}+2916\,{A}^{3}{B}^{9}{D}^{2}-246456\,{A}^{3}{B}^{7}{C}^{3}D+5903280\,{A}^{3}{B}^{5}{C}^{6}-771876\,{A}^{2}{B}^{9}{C}^{2}D+6724944\,{A}^{2}{B}^{7}{C}^{5}$
$\displaystyle-474336\,A{B}^{11}CD+3446064\,A{B}^{9}{C}^{4}-93312\,{B}^{13}D+657072\,{B}^{11}{C}^{3}-260496\,{A}^{5}{B}^{2}{C}^{4}{D}^{2}+82944\,{A}^{5}{C}^{7}D-972000\,{A}^{4}{B}^{4}{C}^{3}{D}^{2}$
$\displaystyle+382320\,{A}^{4}{B}^{2}{C}^{6}D+165888\,{A}^{4}{C}^{9}-1404054\,{A}^{3}{B}^{6}{C}^{2}{D}^{2}-1235736\,{A}^{3}{B}^{4}{C}^{5}D+2744928\,{A}^{3}{C}^{8}{B}^{2}-896184\,{A}^{2}{B}^{8}C{D}^{2}-3742848\,{A}^{2}{B}^{6}{C}^{4}D$
$\displaystyle+6757344\,{A}^{2}{B}^{4}{C}^{7}-209952\,A{B}^{10}{D}^{2}-2978856\,A{B}^{8}{C}^{3}D+5632416\,A{B}^{6}{C}^{6}-751680\,{B}^{10}{C}^{2}D+1531872\,{B}^{8}{C}^{5}-229392\,{A}^{4}{B}^{3}{C}^{2}{D}^{3}$
$\displaystyle-1492992\,{A}^{4}B{C}^{5}{D}^{2}-359397\,{A}^{3}{B}^{5}C{D}^{3}-5794092\,{A}^{3}{B}^{3}{C}^{4}{D}^{2}-497664\,{A}^{3}B{C}^{7}D-139968\,{A}^{2}{B}^{7}{D}^{3}-7986924\,{A}^{2}{B}^{5}{C}^{3}{D}^{2}$
$\displaystyle-4210704\,{A}^{2}{B}^{3}{C}^{6}D+1866240\,{A}^{2}B{C}^{9}-4537296\,A{B}^{7}{C}^{2}{D}^{2}-5668704\,A{B}^{5}{C}^{5}D+3779136\,A{B}^{3}{C}^{8}-886464\,{B}^{9}C{D}^{2}-2072304\,{B}^{7}{C}^{4}D$
$\displaystyle+1726272\,{B}^{5}{C}^{7}-139968\,{A}^{4}{C}^{4}{D}^{3}-6561\,{A}^{3}{B}^{4}{D}^{4}-708588\,{A}^{3}{B}^{2}{C}^{3}{D}^{3}-1819584\,{A}^{3}{C}^{6}{D}^{2}+1010394\,{A}^{2}{B}^{4}{C}^{2}{D}^{3}-8275608\,{A}^{2}{B}^{2}{C}^{5}{D}^{2}$
$\displaystyle-419904\,{A}^{2}{C}^{8}D+2239488\,A{B}^{6}C{D}^{3}-9710280\,A{B}^{4}{C}^{4}{D}^{2}-2729376\,A{B}^{2}{C}^{7}D+653184\,{C}^{10}A+839808\,{B}^{8}{D}^{3}-3289248\,{B}^{6}{C}^{3}{D}^{2}$
$\displaystyle-1959552\,{B}^{4}{C}^{6}D+839808\,{C}^{9}{B}^{2}+1399680\,{A}^{3}B{C}^{2}{D}^{4}+2464749\,{A}^{2}{B}^{3}C{D}^{4}+1994544\,{A}^{2}B{C}^{4}{D}^{3}+1102248\,A{B}^{5}{D}^{4}+7269588\,A{B}^{3}{C}^{3}{D}^{3}$
$\displaystyle-4304016\,AB{C}^{6}{D}^{2}+4199040\,{B}^{5}{C}^{2}{D}^{3}-3464208\,{B}^{3}{C}^{5}{D}^{2}-419904\,B{C}^{8}D+314928\,{A}^{2}{B}^{2}{D}^{5}+3569184\,{A}^{2}{C}^{3}{D}^{4}+5616216\,A{B}^{2}{C}^{2}{D}^{4}$
$\displaystyle+3044304\,A{C}^{5}{D}^{3}+1784592\,{B}^{4}C{D}^{4}+5196312\,{B}^{2}{C}^{4}{D}^{3}-1889568\,{C}^{7}{D}^{2}-472392\,ABC{D}^{5}-1889568\,{B}^{3}{D}^{5}+1495908\,B{C}^{3}{D}^{4}$
$\displaystyle-1889568\,A{D}^{6}-2834352\,{C}^{2}{D}^{5},$
$\displaystyle\epsilon_{1}$
$\displaystyle=-1024\,{A}^{6}{B}^{4}{C}^{4}-3200\,{A}^{5}{B}^{6}{C}^{3}-3744\,{A}^{4}{B}^{8}{C}^{2}-1944\,{A}^{3}{B}^{10}C-378\,{A}^{2}{B}^{12}+1536\,{A}^{5}{C}^{5}{B}^{3}+22848\,{A}^{4}{C}^{4}{B}^{5}+58944\,{A}^{3}{B}^{7}{C}^{3}$
$\displaystyle+61668\,{B}^{9}{C}^{2}{A}^{2}+29160\,A{B}^{11}C+5184\,{B}^{13}+10368\,{A}^{5}{B}^{2}{C}^{4}D+40176\,{A}^{4}{B}^{4}{C}^{3}D+20736\,{A}^{4}{C}^{6}{B}^{2}+56484\,{A}^{3}{B}^{6}{C}^{2}D+124704\,{A}^{3}{C}^{5}{B}^{4}$
$\displaystyle+34506\,{A}^{2}{B}^{8}CD+211464\,{A}^{2}{C}^{4}{B}^{6}+7776\,A{B}^{10}D+141264\,A{B}^{8}{C}^{3}+33048\,{B}^{10}{C}^{2}+9504\,{A}^{4}{B}^{3}{C}^{2}{D}^{2}+57024\,{A}^{4}B{C}^{5}D+14580\,{A}^{3}{B}^{5}C{D}^{2}$
$\displaystyle+220968\,{A}^{3}{B}^{3}{C}^{4}D+38016\,{A}^{3}{C}^{7}B+5589\,{A}^{2}{B}^{7}{D}^{2}+291600\,{A}^{2}{B}^{5}{C}^{3}D+183600\,{A}^{2}{C}^{6}{B}^{3}+160056\,A{B}^{7}{C}^{2}D+215136\,A{C}^{5}{B}^{5}+31104\,{B}^{9}CD$
$\displaystyle+73872\,{B}^{7}{C}^{4}+5184\,{A}^{4}{C}^{4}{D}^{2}-32076\,{A}^{3}{B}^{2}{C}^{3}{D}^{2}+62208\,{A}^{3}{C}^{6}D-199746\,{A}^{2}{B}^{4}{C}^{2}{D}^{2}+274104\,{A}^{2}{B}^{2}{C}^{5}D+20736\,{C}^{8}{A}^{2}-223236\,A{B}^{6}C{D}^{2}$
$\displaystyle+291600\,A{B}^{4}{C}^{4}D+98496\,{C}^{7}A{B}^{2}-69984\,{B}^{8}{D}^{2}+91368\,{B}^{6}{C}^{3}D+64800\,{C}^{6}{B}^{4}-108864\,{A}^{3}B{C}^{2}{D}^{3}-190755\,{A}^{2}{B}^{3}C{D}^{3}-244944\,{A}^{2}B{C}^{4}{D}^{2}$
$\displaystyle-81648\,A{B}^{5}{D}^{3}-612360\,A{B}^{3}{C}^{3}{D}^{2}+139968\,AB{C}^{6}D-312012\,{B}^{5}{C}^{2}{D}^{2}+93312\,{B}^{3}{C}^{5}D+15552\,{C}^{8}B-17496\,{A}^{2}{B}^{2}{D}^{4}-256608\,{A}^{2}{C}^{3}{D}^{3}$
$\displaystyle-352836\,A{B}^{2}{C}^{2}{D}^{3}-221616\,A{C}^{5}{D}^{2}-93312\,{B}^{4}C{D}^{3}-332424\,{B}^{2}{C}^{4}{D}^{2}+69984\,{C}^{7}D+157464\,ABC{D}^{4}$
$\displaystyle+209952\,{B}^{3}{D}^{4}-8748\,B{C}^{3}{D}^{3}+209952\,A{D}^{5}+314928\,{C}^{2}{D}^{4},$
$\displaystyle\delta_{0}$
$\displaystyle=10240\,{A}^{8}{B}^{6}{C}^{5}+39936\,{A}^{7}{B}^{8}{C}^{4}+62208\,{A}^{6}{B}^{10}{C}^{3}+48384\,{A}^{5}{B}^{12}{C}^{2}+18792\,{A}^{4}{B}^{14}C+2916\,{A}^{3}{B}^{16}+98304\,{A}^{8}{B}^{3}{C}^{7}+580608\,{A}^{7}{B}^{5}{C}^{6}$
$\displaystyle+1208064\,{A}^{6}{B}^{7}{C}^{5}+1082688\,{A}^{5}{B}^{9}{C}^{4}+267984\,{A}^{4}{B}^{11}{C}^{3}-220320\,{A}^{3}{B}^{13}{C}^{2}-162648\,{A}^{2}{B}^{15}C-31104\,A{B}^{17}+172800\,{A}^{7}{B}^{4}{C}^{5}D$
$\displaystyle+442368\,{A}^{7}{C}^{8}{B}^{2}+620928\,{A}^{6}{B}^{6}{C}^{4}D+1110528\,{A}^{6}{C}^{7}{B}^{4}+895536\,{A}^{5}{B}^{8}{C}^{3}D-2439936\,{A}^{5}{C}^{6}{B}^{6}+649296\,{A}^{4}{B}^{10}{C}^{2}D-11284704\,{A}^{4}{C}^{5}{B}^{8}$
$\displaystyle+237168\,{A}^{3}{B}^{12}CD-15591456\,{A}^{3}{C}^{4}{B}^{10}+34992\,{A}^{2}{B}^{14}D-10362816\,{A}^{2}{C}^{3}{B}^{12}-3386448\,A{B}^{14}{C}^{2}-435456\,{B}^{16}C+331776\,{A}^{7}B{C}^{7}D$
$\displaystyle-67392\,{A}^{6}{B}^{5}{C}^{3}{D}^{2}+4648320\,{A}^{6}{B}^{3}{C}^{6}D+663552\,{A}^{6}{C}^{9}B-147744\,{A}^{5}{B}^{7}{C}^{2}{D}^{2}+14486688\,{A}^{5}{B}^{5}{C}^{5}D-3103488\,{A}^{5}{C}^{8}{B}^{3}$
$\displaystyle-107892\,{A}^{4}{B}^{9}C{D}^{2}+21505824\,{A}^{4}{B}^{7}{C}^{4}D-25468992\,{A}^{4}{C}^{7}{B}^{5}-26244\,{A}^{3}{B}^{11}{D}^{2}+18424800\,{A}^{3}{B}^{9}{C}^{3}D-52265088\,{A}^{3}{C}^{6}{B}^{7}$
$\displaystyle+9572256\,{A}^{2}{B}^{11}{C}^{2}D-47708352\,{A}^{2}{C}^{5}{B}^{9}+2846016\,A{B}^{13}CD-20497536\,A{C}^{4}{B}^{11}+373248\,{B}^{15}D-3382560\,{B}^{13}{C}^{3}-2519424\,{A}^{6}{B}^{2}{C}^{5}{D}^{2}$
$\displaystyle+497664\,{A}^{6}{C}^{8}D-7235568\,{A}^{5}{B}^{4}{C}^{4}{D}^{2}+12845952\,{A}^{5}{B}^{2}{C}^{7}D+331776\,{A}^{5}{C}^{10}-5445144\,{A}^{4}{B}^{6}{C}^{3}{D}^{2}+31290624\,{A}^{4}{B}^{4}{C}^{6}D$
$\displaystyle-10886400\,{A}^{4}{C}^{9}{B}^{2}+1279152\,{A}^{3}{B}^{8}{C}^{2}{D}^{2}+31648320\,{A}^{3}{B}^{6}{C}^{5}D-50258880\,{A}^{3}{C}^{8}{B}^{4}+2886840\,{A}^{2}{B}^{10}C{D}^{2}+17184960\,{A}^{2}{B}^{8}{C}^{4}D$
$\displaystyle-74442240\,{A}^{2}{C}^{7}{B}^{6}+839808\,A{B}^{12}{D}^{2}+5520960\,A{B}^{10}{C}^{3}D-45567360\,A{C}^{6}{B}^{8}+886464\,{B}^{12}{C}^{2}D-9984384\,{B}^{10}{C}^{5}-925344\,{A}^{5}{B}^{3}{C}^{3}{D}^{3}$
$\displaystyle-6718464\,{A}^{5}B{C}^{6}{D}^{2}-1924560\,{A}^{4}{B}^{5}{C}^{2}{D}^{3}+23001408\,{A}^{4}{B}^{3}{C}^{5}{D}^{2}+15676416\,{A}^{4}B{C}^{8}D-1347192\,{A}^{3}{B}^{7}C{D}^{3}+116301744\,{A}^{3}{B}^{5}{C}^{4}{D}^{2}$
$\displaystyle+9859968\,{A}^{3}{B}^{3}{C}^{7}D-10948608\,{A}^{3}{C}^{10}B-314928\,{A}^{2}{B}^{9}{D}^{3}+148785984\,{A}^{2}{B}^{7}{C}^{3}{D}^{2}-20435328\,{A}^{2}{B}^{5}{C}^{6}D-40217472\,{A}^{2}{C}^{9}{B}^{3}$
$\displaystyle+76667472\,A{B}^{9}{C}^{2}{D}^{2}-20808576\,A{B}^{7}{C}^{5}D-42177024\,A{C}^{8}{B}^{5}+13996800\,{B}^{11}C{D}^{2}-4945536\,{B}^{9}{C}^{4}D-13561344\,{B}^{7}{C}^{7}-1119744\,{A}^{5}{C}^{5}{D}^{3}$
$\displaystyle+104976\,{A}^{4}{B}^{4}C{D}^{4}-36181728\,{A}^{4}{B}^{2}{C}^{4}{D}^{3}-3359232\,{A}^{4}{C}^{7}{D}^{2}+59049\,{A}^{3}{B}^{6}{D}^{4}-96799536\,{A}^{3}{B}^{4}{C}^{3}{D}^{3}+145006848\,{A}^{3}{B}^{2}{C}^{6}{D}^{2}$
$\displaystyle+8584704\,{A}^{3}{C}^{9}D-104941008\,{A}^{2}{B}^{6}{C}^{2}{D}^{3}+380642976\,{A}^{2}{B}^{4}{C}^{5}{D}^{2}-40310784\,{A}^{2}{B}^{2}{C}^{8}D-3732480\,{C}^{11}{A}^{2}-52488000\,A{B}^{8}C{D}^{3}$
$\displaystyle+311148864\,A{B}^{6}{C}^{4}{D}^{2}-67184640\,A{B}^{4}{C}^{7}D-12877056\,{C}^{10}A{B}^{2}-10077696\,{B}^{10}{D}^{3}+81671328\,{B}^{8}{C}^{3}{D}^{2}-23887872\,{B}^{6}{C}^{6}D$
$\displaystyle-7838208\,{B}^{4}{C}^{9}+11757312\,{A}^{4}B{C}^{3}{D}^{4}+11354904\,{A}^{3}{B}^{3}{C}^{2}{D}^{4}-115893504\,{A}^{3}B{C}^{5}{D}^{3}-6337926\,{A}^{2}{B}^{5}C{D}^{4}-257191200\,{A}^{2}{B}^{3}{C}^{4}{D}^{3}$
$\displaystyle+193155840\,{A}^{2}B{C}^{7}{D}^{2}-5668704\,A{B}^{7}{D}^{4}-203443488\,A{B}^{5}{C}^{3}{D}^{3}+353139264\,A{B}^{3}{C}^{6}{D}^{2}-47029248\,AB{C}^{9}D-55007424\,{B}^{7}{C}^{2}{D}^{3}$
$\displaystyle+151585344\,{B}^{5}{C}^{5}{D}^{2}-33592320\,{B}^{3}{C}^{8}D-1119744\,{C}^{11}B+1259712\,{A}^{3}{B}^{2}C{D}^{5}+15116544\,{A}^{3}{C}^{4}{D}^{4}+708588\,{A}^{2}{B}^{4}{D}^{5}$
$\displaystyle-151795296\,{A}^{2}{B}^{2}{C}^{3}{D}^{4}-116733312\,{A}^{2}{C}^{6}{D}^{3}-251233812\,A{B}^{4}{C}^{2}{D}^{4}-159983424\,A{B}^{2}{C}^{5}{D}^{3}+85660416\,A{C}^{8}{D}^{2}-89439552\,{B}^{6}C{D}^{4}$
$\displaystyle-71803584\,{B}^{4}{C}^{4}{D}^{3}+95738112\,{B}^{2}{C}^{7}{D}^{2}-13436928\,{C}^{10}D+117153216\,{A}^{2}B{C}^{2}{D}^{5}+178564176\,A{B}^{3}C{D}^{5}-319336992\,AB{C}^{4}{D}^{4}$
$\displaystyle+68024448\,{B}^{5}{D}^{5}-268791048\,{B}^{3}{C}^{3}{D}^{4}+42830208\,B{C}^{6}{D}^{3}-15116544\,{A}^{2}C{D}^{6}+11337408\,A{B}^{2}{D}^{6}+249422976\,A{C}^{3}{D}^{5}$
$\displaystyle+229582512\,{B}^{2}{C}^{2}{D}^{5}-120932352\,{C}^{5}{D}^{4}+158723712\,BC{D}^{6}-136048896\,{D}^{7},$
$\displaystyle\epsilon_{0}$
$\displaystyle=128\,{A}^{6}{B}^{5}{C}^{3}+288\,{A}^{5}{B}^{7}{C}^{2}+216\,{A}^{4}{B}^{9}C+54\,{A}^{3}{B}^{11}+6144\,{A}^{6}{B}^{2}{C}^{5}+22272\,{A}^{5}{B}^{4}{C}^{4}+28992\,{A}^{4}{B}^{6}{C}^{3}+16272\,{A}^{3}{B}^{8}{C}^{2}+3348\,{A}^{2}{B}^{10}C$
$\displaystyle+48\,{A}^{5}{B}^{3}{C}^{3}D+18432\,{A}^{5}{C}^{6}B+468\,{A}^{4}{B}^{5}{C}^{2}D+6048\,{A}^{4}{C}^{5}{B}^{3}+756\,{A}^{3}{B}^{7}CD-108936\,{A}^{3}{C}^{4}{B}^{5}+324\,{A}^{2}{B}^{9}D-178992\,{A}^{2}{C}^{3}{B}^{7}$
$\displaystyle-103896\,A{C}^{2}{B}^{9}-20736\,C{B}^{11}+2304\,{A}^{5}{C}^{5}D-324\,{A}^{4}{B}^{4}C{D}^{2}+73872\,{A}^{4}{B}^{2}{C}^{4}D+13824\,{A}^{4}{C}^{7}-243\,{A}^{3}{B}^{6}{D}^{2}+237096\,{A}^{3}{B}^{4}{C}^{3}D$
$\displaystyle-151200\,{A}^{3}{C}^{6}{B}^{2}+292392\,{A}^{2}{B}^{6}{C}^{2}D-460080\,{A}^{2}{C}^{5}{B}^{4}+157680\,A{B}^{8}CD-400896\,A{C}^{4}{B}^{6}+31104\,{B}^{10}D-110160\,{C}^{3}{B}^{8}-51840\,{A}^{4}B{C}^{3}{D}^{2}$
$\displaystyle-88020\,{A}^{3}{B}^{3}{C}^{2}{D}^{2}+235008\,{A}^{3}B{C}^{5}D-36450\,{A}^{2}{B}^{5}C{D}^{2}+648000\,{A}^{2}{B}^{3}{C}^{4}D-228096\,{A}^{2}{C}^{7}B+589680\,A{B}^{5}{C}^{3}D-432864\,A{C}^{6}{B}^{3}$
$\displaystyle+174096\,{B}^{7}{C}^{2}D-191808\,{B}^{5}{C}^{5}-72576\,{A}^{3}{C}^{4}{D}^{2}-1458\,{A}^{2}{B}^{4}{D}^{3}+326592\,{A}^{2}{B}^{2}{C}^{3}{D}^{2}+212544\,{A}^{2}{C}^{6}D+615276\,A{B}^{4}{C}^{2}{D}^{2}$
$\displaystyle+427680\,A{B}^{2}{C}^{5}D-93312\,A{C}^{8}+238464\,{B}^{6}C{D}^{2}+225504\,{B}^{4}{C}^{4}D-108864\,{B}^{2}{C}^{7}-489888\,{A}^{2}B{C}^{2}{D}^{3}-833976\,A{B}^{3}C{D}^{3}+828144\,AB{C}^{4}{D}^{2}$
$\displaystyle-326592\,{B}^{5}{D}^{3}+659016\,{B}^{3}{C}^{3}{D}^{2}-15552\,B{C}^{6}D+93312\,{A}^{2}C{D}^{4}-979776\,A{C}^{3}{D}^{3}-1032264\,{B}^{2}{C}^{2}{D}^{3}+373248\,{C}^{5}{D}^{2}-559872\,BC{D}^{4}$
$\displaystyle+839808\,{D}^{5}.$

## References

  * [1] Oskar Bolza, _On binary sextics with linear transformations into themselves_ , Amer. J. Math. 10 (1887), no. 1, 47–70. MR 1505464
  * [2] A. Clebsch and P. Gordan, _Theorie der Abelschen Functionen_ , Physica-Verlag, Würzburg, 1967, Thesaurus Mathematicae, (Neudrucke wichtiger mathematischer Werke), Band 7. MR 0485106
  * [3] E. Freitag, _Siegelsche Modulfunktionen_ , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 254, Springer-Verlag, Berlin, 1983. MR 871067
  * [4] V. A. Gritsenko and V. V. Nikulin, _Igusa modular forms and “the simplest” Lorentzian Kac-Moody algebras_ , Mat. Sb. 187 (1996), no. 11, 27–66. MR 1438983
  * [5] Jun-Ichi Igusa, _Arithmetic variety of moduli for genus two_ , Ann. of Math. (2) 72 (1960), 612–649. MR 0114819
  * [6] by same author, _On Siegel modular forms of genus two_ , Amer. J. Math. 84 (1962), 175–200. MR 0141643
  * [7] by same author, _Modular forms and projective invariants_ , Amer. J. Math. 89 (1967), 817–855. MR 0229643
  * [8] by same author, _On the ring of modular forms of degree two over ${\bf Z}$_, Amer. J. Math. 101 (1979), no. 1, 149–183. MR 527830
  * [9] Vishwanath Krishnamoorthy, Tanush Shaska, and Helmut Voelklein, _Invariants of binary forms_ , Progress in Galois theory, Dev. Math., vol. 12, Springer, New York, 2005, pp. 101–122. MR 2148462
  * [10] Andreas Malmendier and David R. Morrison, _K3 surfaces, modular forms, and non-geometric heterotic compactifications_ , Lett. Math. Phys. 105 (2015), no. 8, 1085–1118. MR 3366121
  * [11] Jean-François Mestre, _Construction de courbes de genre $2$ à partir de leurs modules_, Effective methods in algebraic geometry (Castiglioncello, 1990), Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 313–334. MR 1106431
  * [12] Tanush Shaska and Helmut Voelklein, _Elliptic subfields and automorphisms of genus 2 function fields_ , Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), Springer, Berlin, 2004, pp. 703–723. MR 2037120


# Soft-Search: Two Datasets to Study the Identification and Production of
Research Software

Eva Maxfield Brown<EMAIL_ADDRESS>0000-0003-2564-0373 University of Washington
Information SchoolSeattleWashingtonUSA ,  Lindsey Schwartz<EMAIL_ADDRESS>0000-0001-8338-0288 University of Washington Information
SchoolSeattleWashingtonUSA ,  Richard Lewei Huang<EMAIL_ADDRESS>0000-0002-0264-9300 University of Washington Information
SchoolSeattleWashingtonUSA and  Nicholas Weber<EMAIL_ADDRESS>0000-0002-6008-3763 University of Washington Information
SchoolSeattleWashingtonUSA

(2018)

###### Abstract.

Software is an important tool for scholarly work, but software produced for
research is in many cases not easily identifiable or discoverable. A potential
first step in linking research and software is software identification. In
this paper we present two datasets to study the identification and production
of research software. The first dataset contains almost 1000 human labeled
annotations of software production from National Science Foundation (NSF)
awarded research projects. We use this dataset to train models that predict
software production. Our second dataset is created by applying the trained
predictive models across the abstracts and project outcomes reports for all
NSF funded projects between the years of 2010 and 2023. The result is an
inferred dataset of software production for over 150,000 NSF awards. We
release the Soft-Search dataset to aid in identifying and understanding
research software production: https://github.com/si2-urssi/eager

datasets, text classification, research software

††copyright: acmcopyright††journalyear: 2018††doi:
XXXXXXX.XXXXXXX††conference: Make sure to enter the correct conference title
from your rights confirmation emai; June 03–05, 2018; Woodstock, NY††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Information systems Digital
libraries and archives††ccs: Information systems Clustering and
classification††ccs: Software and its engineering Software creation and
management††ccs: Software and its engineering Software libraries and
repositories

##  1\. Introduction

Software production, use, and reuse is an increasingly crucial part of
scholarly work (Bhattarai et al., 2022; Trisovic et al., 2021). While
historically underutilized, citing and referencing software used during the
course of research is becoming common with new standards for software citation
(Katz et al., 2021; Du et al., 2022) and work in extracting software
references in existing literature (Istrate et al., 2022). However, records of
software production are not readily identifiable or available at scale in the
way that peer-reviewed publications or other scholarly outputs are (Schindler
et al., 2022). To make progress on this problem, we introduce two related
datasets for studying and inferring software produced as a part of research,
which we refer to as the Soft-Search dataset.

The Soft-Search dataset is aimed at identifying research projects which are
likely to have produced software while funded by a federal grant. We start by
identifying GitHub repositories that acknowledge funding from at least one
National Science Foundation (NSF) award. We then annotate each GitHub
repository found with a binary decision for its contents: software or not-
software (e.g. not all github repositories contain software, they might
include research notes, course materials, etc.). We then link each annotated
GitHub repository to the specific NSF award ID(s) referenced in its README.md
file. Finally, we compile the Soft-Search Training dataset using the
annotations for each GitHub repository, and the text from the linked NSF award
abstract and the project outcomes report.

Using the Soft-Search Training dataset, we train a variety of models to
predict software production using either the NSF award abstract or project
outcomes report text as input. We use the best performing models to then infer
software production against all awards funded by the National Science
Foundation from 2010 to 2023 (additional details are offered in Section 2).
The predictions and metadata for each NSF award between the 2010 and 2023
period are compiled to form the Soft-Search Inferred dataset.

In total, our new Soft-Search dataset includes the following contributions:

  1. 1.

Soft-Search Training: A ground truth dataset compiled using linked NSF awards
and GitHub repositories which have been annotated for software production.

  2. 2.

Multiple classifiers which infer software production from either the text of
an NSF award’s abstract or project outcomes report.

  3. 3.

Soft-Search Inferred: A dataset of more than 150,000 NSF funded awards from
between 2010 and 2023. Each award has two predictions for software production:
one from prediction using the abstract text and the other from prediction
using the project outcomes report text.

The rest of the paper proceeds as follows. In Section 2 we detail the data
collection and annotation process used for creating the Soft-Search Training
dataset. In Section 3 we briefly describe the model training process and
report results. In Section 4 we provide summary statistics for the Soft-Search
Inferred dataset and observe trends in software production over time. We
conclude with discussion regarding the limitations of our approach and
opportunities for future work.

##  2\. Data Collection and Annotation

###  2.1. Finding Software Produced by NSF Awards

The first step in our data collection process was to find software outputs
from National Science Foundation (NSF) funded research. This step has two
potential approaches. The first approach is a manual search for references and
promises of software production within NSF award abstracts, project outcome
reports, and papers supported by each award. This first approach is labor
intensive and may be prone to labeling errors because while there may be a
promise of software production in these documents, it may not be possible to
verify such software was ultimately produced. The other approach is to predict
software production using a trained model. We pursue this approach with the
caveat that there are also potential label errors.

To gather examples of verifiable software production, we created a Python
script which used the GitHub API to search for repositories which included
reference to financial support from an NSF award in the repositories README.md
file. Specifically our script queried for README.md files which contained any
of the following text snippets: ‘National Science Foundation’, ‘NSF Award’,
‘NSF Grant’, ‘Supported by NSF’, or ‘Supported by the NSF’. GitHub was
selected as the basis for our search because of its widespread adoption and
mention in scholarly publication (Escamilla et al., 2022). This search found
1520 unique repositories which contained a reference to the NSF in the
repository’s README.md file.

###  2.2. Software Production Annotation

The next step in our data collection process was to annotate each of the
GitHub repositories found as either “software” or “not software.” In our
initial review of the repositories we had collected, we found that the content
of repositories ranged from documentation, experimental notes, course
materials, collections of one-off scripts written during a research project,
to more typical software libraries with installation instructions, testing,
and community support and use.

Using existing definitions of what constitutes research software to form the
basis of our annotation criteria (Martinez-Ortiz et al., 2022; Sochat et al.,
2022), we conducted multiple rounds of trial coding on samples of the data.
Fleiss’ kappa was used to determine if there was agreement between our
research team on whether ten GitHub repositories contained ‘software’ or not.
On each round of trial coding ten GitHub repositories were randomly selected
from our dataset for each member of our research team to annotate
independently. When assessing a repository, members of the research team were
allowed to use any information in the repository to determine their annotation
(i.e. the content of the README.md file, the repository activity,
documentation availability, etc.)

Our final round of trial coding showed that there was near perfect agreement
between the research team (K=0.892) (Viera et al., 2005).

Our final annotation criteria was generally inclusive of labeling repositories
as software, rather there were specific exclusion criteria that resulted in a
repository being labeled as “not software”. Specifically repositories were
labeled as “not software” when a repository primarily consisted of:

  1. 1.

project documentation or research notes

  2. 2.

teaching materials for a workshop or course

  3. 3.

the source code for a project or research lab website

  4. 4.

collections of scripts specific to the analysis of a single experiment without
regard to further generalizability

  5. 5.

utility functions for accessing data without providing any additional
processing capacity

We then annotated all GitHub repositories in our dataset as either “software”
or “not software” according to our agreed upon annotation criteria.

###  2.3. Linking GitHub Repositories to NSF Awards

Our final step in the data collection process was to link the annotated GitHub
repositories back to specific NSF awards. To do so, we created a script which
would load the webpage for each GitHub repository, scrape the content of the
repository’s README and find the specific NSF award ID number(s) referenced.
While annotating the dataset, and with this script, our dataset size was
reduced as we found some repositories were returned in the initial search
because of the “NSF” acronym being used by other, non-United-States
governmental agencies which also fund research.

When processing each repository, our Python script would load the README
content, search for NSF Award ID patterns with regular expressions, and then
verify that each NSF award ID found was valid by requesting metadata for the
award from the NSF award API.

We then retrieved the text for each award’s abstract and project outcomes
report. This was the final step of our data collection process and allowed us
to create a dataset of 446 unique NSF awards labeled as ‘produced software’
and 471 unique NSF awards labeled as ‘did not produce software’.

##  3\. Predictive Models

Using the compiled Soft-Search Training dataset, we trained three different
models using the text from either the award abstract or project outcomes
report. The models trained include a logistic regression model trained with
TF-IDF word embeddings (tfidf-logit), a logistic regression model trained with
semantic embeddings (semantic-logit), and a fine-tuned transformer
(transformer). The semantic embeddings and the base model from which we fine-
tuned our own transformer model was the ‘distilbert-base-uncased-finetuned-
sst-2-english’ model (HF Canonical Model Maintainers, 2022). Each model was
trained with 80% of the Soft-Search Training dataset. We then test each of the
models and use F1 to rank each model’s performance.

Table 1. Predictive Model Results (Trained with Abstract Text)  model accuracy
precision recall f1 0 tfidf-logit 0.674 0.674 0.674 0.673 1 transformer 0.636
0.608 0.697 0.649 2 semantic-logit 0.630 0.630 0.630 0.630 3 regex 0.516 0.515
0.516 0.514

Table 1 reports the results from training using the abstract text as input.
The best performing model was the tfidf-logit which achieved an F1 of 0.673.

Table 2. Predictive Model Results (Trained with Project Outcomes Report Text)  | model | accuracy | precision | recall | f1
---|---|---|---|---|---
0 | tfidf-logit | 0.745 | 0.745 | 0.745 | 0.745
1 | transformer | 0.673 | 0.638 | 0.771 | 0.698
2 | semantic-logit | 0.633 | 0.633 | 0.633 | 0.632
3 | regex | 0.510 | 0.507 | 0.510 | 0.482

Table 2 reports the results from training using the project outcomes reports
as input. The best performing model was the tfidf-logit which achieved an F1
of 0.745.

While the models trained with the project outcomes reports were trained with
less data, the best model of the group achieved a higher F1 than any of the
models trained with the abstracts. While we have not investigated further, we
believe this to be because the project outcomes reports contain more direct
citation of produced software rather than an abstract’s promise of software
production.

The data used for training, and functions to reproduce these models, are made
available via our Python package: soft-search.

##  4\. The Soft-Search Dataset

Using the predictive models, we compile the Soft-Search Inferred dataset which
contains the metadata, abstract text, and project outcomes report text, for
all NSF awarded projects during the 2010-2023 period. The Soft-Search Inferred
dataset additionally contains our predictions for software production using
both texts respectively.

Table 3. Composition of the NSF Soft Search Dataset  | Program | # Awards | # Software | % Software
---|---|---|---|---
0 | MPS | 32885 | 19178 | 0.583184
1 | CISE | 24633 | 13274 | 0.538871
2 | ENG | 22900 | 11242 | 0.490917
3 | GEO | 17822 | 5142 | 0.288520
4 | BIO | 16990 | 6013 | 0.353914
5 | EHR | 13703 | 575 | 0.041962
6 | SBE | 13318 | 1966 | 0.147620
7 | TIP | 8597 | 4501 | 0.523555
8 | OISE | 2329 | 636 | 0.273079
9 | OIA | 498 | 123 | 0.246988

###  4.1. Trends and Observations

Figure 1. Software Production Over Time (Using Predictions from Abstracts)

Using the Soft-Search Inferred dataset we can observe trends in software
production over time. Figure 1 plots the percent of awards which we predict to
have produced software (using the award’s abstract) over time. While there are
minor year-to-year deviations in predicted software production, we observe the
“Math and Physical Sciences” (MPS) funding program as funding the most awards
which we predict to produce software, with “Computer and Information Science
and Engineering” (CISE), and “Engineering” (ENG) close behind.

Figure 2. Software Production Grouped By Award Duration (Using Predictions
from Abstracts)

We can additionally observe trends in software production as award duration
increases. Figure 2 plots the percent of awards which we predict to have
produced software (using the award’s abstract) grouped by the award duration
in years. We note that as award duration increases, the percentage of awards
which are predicted to have produced software also tends to increase.

##  5\. Conclusion

We introduce Soft-Search, a pair of novel datasets for studying software
production from NSF funded projects. The Soft-Search Training dataset is a
human-labeled dataset with almost 1000 examples used to train models which
predict software production from either the NSF award abstract text or the
project outcomes report text. We used these models to generate the Soft-Search
Inferred dataset. The Soft-Search Inferred dataset includes project metadata,
the awards abstract and project outcomes report, and predictions of software
production for each NSF funded project between 2010 and 2023. We hope that
Soft-Search helps further new studies and findings in understanding the role
software development plays in scholarly publication.

All datasets and predictive models produced by this work are available from
our GitHub repository: si2-urssi/eager.

###  5.1. Limitations

As discussed in Section 2, the Soft-Search Training dataset was entirely
composed of NSF awards which ultimately released or hosted software (and other
research products) on GitHub. Due to our data collection strategy, it is
possible that each of the predictive models learned not to predict if an NSF
award would produce software, but rather, if an NSF award would produce
software hosted on GitHub.

###  5.2. Future Work

As discussed in Section 2.1, our initial method for attempting to find
research software produced from NSF supported awards was to search for
references and promises of software production in the abstract, project
outcomes report, and attached papers of each award. While attempting this
approach to create the dataset, we found that many awards and papers that
reference computational methods do not provide a reference web link to their
code repositories or websites. In some cases, we found repositories related to
an award or paper via Google and GitHub search ourselves. While we support
including references to code repositories in award abstracts, outcomes
reports, and papers, future research should be conducted on how to enable
automatic reconnection of papers and their software outputs.

##  6\. Acknowledgements

We thank the USRSSI team, especially Karthik Ram for their input. This
material is based upon work supported by the National Science Foundation under
Grant 2211275.

## References

  * (1)
  * Bhattarai et al. (2022) Prajjwal Bhattarai, Mohammed Ghassemi, and Tuka Alhanai. 2022\.  Open-Source Code Repository Attributes Predict Impact of Computer Science Research. In _Proceedings of the 22nd ACM/IEEE Joint Conference on Digital Libraries_ (Cologne, Germany) _(JCDL ’22)_. Association for Computing Machinery, New York, NY, USA, Article 16, 7 pages.  https://doi.org/10.1145/3529372.3530927
  * Du et al. (2022) Cai Fan Du, Johanna Cohoon, Patrice Lopez, and James Howison. 2022.  Understanding progress in software citation: a study of software citation in the CORD-19 corpus.  _PeerJ Computer Science_ 8 (2022).
  * Escamilla et al. (2022) Emily Escamilla, Martin Klein, Talya Cooper, Vicky Rampin, Michele C. Weigle, and Michael L. Nelson. 2022.  The Rise of GitHub in Scholarly Publications.  https://doi.org/10.48550/ARXIV.2208.04895
  * HF Canonical Model Maintainers (2022) HF Canonical Model Maintainers. 2022\.  distilbert-base-uncased-finetuned-sst-2-english (Revision bfdd146).  https://doi.org/10.57967/hf/0181
  * Istrate et al. (2022) Ana-Maria Istrate, Donghui Li, Dario Taraborelli, Michaela Torkar, Boris Veytsman, and Ivana Williams. 2022\.  A large dataset of software mentions in the biomedical literature.  https://doi.org/10.48550/ARXIV.2209.00693
  * Katz et al. (2021) Daniel S. Katz, Neil P. Chue Hong, Tim Clark, August Muench, Shelley Stall, Daina R. Bouquin, Matthew Cannon, Scott C. Edmunds, Telli Faez, Patricia Feeney, Martin Fenner, Michael Friedman, Gerry Grenier, Melissa Harrison, Joerg Heber, Adam Leary, Catriona J. MacCallum, Hollydawn Murray, Erika Pastrana, Kath Perry, Douglas C. Schuster, Martina Stockhause, and Jake S. Yeston. 2021.  Recognizing the value of software: a software citation guide.  _F1000Research_ 9 (2021).
  * Martinez-Ortiz et al. (2022) Carlos Martinez-Ortiz, Paula Martinez Lavanchy, Laurents Sesink, Brett G. Olivier, James Meakin, Maaike de Jong, and Maria Cruz. 2022\.  Practical guide to Software Management Plans.  https://doi.org/10.5281/zenodo.7185371
  * Schindler et al. (2022) David Schindler, Felix Bensmann, Stefan Dietze, and Frank Krüger. 2022.  The role of software in science: a knowledge graph-based analysis of software mentions in PubMed Central.  _PeerJ Computer Science_ 8 (2022).
  * Sochat et al. (2022) Vanessa Sochat, Nicholas May, Ian Cosden, Carlos Martinez-Ortiz, and Sadie Bartholomew. 2022.  The Research Software Encyclopedia: a community framework to define research software.  _Journal of Open Research Software_ (2022).
  * Trisovic et al. (2021) Ana Trisovic, Matthew K. Lau, Thomas Pasquier, and Mercè Crosas. 2021.  A large-scale study on research code quality and execution.  _Scientific Data_ 9 (2021).
  * Viera et al. (2005) Anthony J Viera, Joanne M Garrett, et al. 2005\.  Understanding interobserver agreement: the kappa statistic.  _Fam med_ 37, 5 (2005), 360–363.