| Curve name |
$X_{195h}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 7 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{195}$ |
| Curves that $X_{195h}$ minimally covers |
|
| Curves that minimally cover $X_{195h}$ |
|
| Curves that minimally cover $X_{195h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -6912t^{16} + 103680t^{12} - 57888t^{8} + 6480t^{4} - 27\]
\[B(t) = -221184t^{24} - 6967296t^{20} + 14390784t^{16} - 6096384t^{12} +
899424t^{8} - 27216t^{4} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 560x + 2900$, with conductor $240$ |
| Generic density of odd order reductions |
$19/336$ |