Discriminant $225$
j-invariant \( \frac{13997521}{225} \)
Torsion structure \(\Z/{2}\Z \times \Z/{4}\Z\)
sage: E = EllipticCurve([1, 1, 1, -5, 2]) # or
gp: E = ellinit([1, 1, 1, -5, 2]) \\ or
magma: E := EllipticCurve([1, 1, 1, -5, 2]); // or
\(y^2+xy+y=x^3+x^2-5x+2\)
\(\Z/{2}\Z \times \Z/{4}\Z\)
\( \left(-3, 1\right) \), \( \left(2, 1\right) \)
\( \left(-3, 1\right) \), \( \left(0, 1\right) \), \( \left(0, -2\right) \), \( \left(1, -1\right) \), \( \left(2, 1\right) \), \( \left(2, -4\right) \)
Discriminant: \(225 \) = \(3^{2} \cdot 5^{2} \)
j-invariant: \( \frac{13997521}{225} \) = \(3^{-2} \cdot 5^{-2} \cdot 241^{3}\)
The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X215c.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 7 & 14 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right)$ and has index 96.
\(2\) Cs
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{4}\Z$ are as follows:
$2$ \(\Q(\sqrt{5}) \) \(\Z/2\Z \times \Z/8\Z\) 2.2.5.1-45.1-a6
$4$ \(\Q(i, \sqrt{15})\) \(\Z/4\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\zeta_{12})\) \(\Z/2\Z \times \Z/8\Z\) Not in database
$4$ \(\Q(\zeta_{15})^+\) \(\Z/2\Z \times \Z/16\Z\) 4.4.1125.1-45.1-b8
$8$ 8.0.12960000.1 \(\Z/4\Z \times \Z/8\Z\) Not in database
$8$ 8.2.110716875.2 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ 16.0.1426576071720960000.2 \(\Z/2\Z \times \Z/16\Z\) Not in database
$16$ \(\Q(\zeta_{60})\) \(\Z/4\Z \times \Z/16\Z\) Not in database