even characteristic is zero. The same structure is observed for higher genera hyperelliptic curves even for $g>5$.

The characteristics entering to the Rosenhain formula are described as follows. Take any of 15 Rosenhain derivative formula, say, $$\theta_1[p] \theta_2[q]-\theta_2[p] \theta_1[q]
= \pi^2\theta[\gamma_1]\theta[\gamma_2]\theta[\gamma_3]\theta[\gamma_4]$$ Then 10 even characteristics can be grouped as $$\underbrace{ [\gamma_1],\ldots,[\gamma_4]}_4, \; \underbrace{ [\alpha_1],[\alpha_2],[\alpha_3] }_{[\alpha_1]+ [\alpha_2]+ [\alpha_3]=[p]},\; \;
\underbrace{ [\beta_1],[\beta_2],[\beta_3] }_{ [\beta_1]+ [\beta_2]+ [\beta_3]=[q]},$$ Then matrix of $\mathfrak{a}$-periods $$\begin{aligned}
\mathcal{A} &= \frac{2Q}{PR} \left( \begin{array}{rr}
Q\theta_1[q]&Q \theta_2[q]\\
P\theta_1[p]& P\theta_2[p]
     \end{array}  \right)\end{aligned}$$ with $$\begin{aligned}
P=\prod_{j=1}^3\theta[\alpha_j], \;\;Q=\prod_{j=1}^3\theta[\beta_j],\; \;R=\prod_{j=1}^4\theta[\gamma_j]\label{PQR}\end{aligned}$$

Note, that the 15 curves are given as $$\begin{aligned}
 \mathcal{C}_{p,q}:   \quad y^2=x(x-1)(x-a_1)(x-a_2)(x-a_3) \label{Bolza}   \end{aligned}$$ where branch points are computed by [*Bolza formulae*]{} [@bol886], $$\begin{aligned}
 e[\delta_j]=  - \frac{\partial_{{U}_1} \theta[\delta_j] }{\partial_{\boldsymbol{U}_2} \theta[\delta_j]}  , \quad
j=1,\ldots,6
\label{bolza1}
  \end{aligned}$$ where $\partial_{{U}_i}$ directional derivative along vector ${U}_i$, $i=1,2$, $\mathcal{A}^{-1}=({U}_1,{U}_2)$ and $$e[p]=0,\; e[q]=\infty,$$ All 15 curves $\mathcal{C}_{p,q}$ are Möbius equivalent.

 
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Generalization of the Rosenhain formula to higher genera hyperelliptic curve was found in [@er08] and developed further in [@eil18].

$$\begin{aligned}
\begin{split} y^2&=\phi(x) \psi(x)\\ \phi(x)&= \prod_{k=1}^g(x-e_{2k}), \;
\psi(x)= \prod_{k=1}^{g+1}(x-e_{2k-1}) \label{curve-g}  \end{split}   \end{aligned}$$

Denote $ R=\prod_{k=1}^{g+2}\theta[\gamma_k]$ monomial in left had side of Riemann-Jacobi formula (\[RiemannJacobi\])

Let genus $g$ hyperelliptic curve is given in (\[curve-g\]). Then winding vectors $(U_1,\ldots,U_g)=\mathcal{A}^{-1}$ are given by the 