| Curve name |
$X_{240b}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 16 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 16 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{240}$ |
| Curves that $X_{240b}$ minimally covers |
|
| Curves that minimally cover $X_{240b}$ |
|
| Curves that minimally cover $X_{240b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{24} + 216t^{20} - 3456t^{12} + 6480t^{8} + 3456t^{4} - 6912\]
\[B(t) = -54t^{36} + 648t^{32} - 1296t^{28} - 12096t^{24} + 55728t^{20} -
5184t^{16} - 314496t^{12} + 456192t^{8} - 165888t^{4} + 221184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 720x - 7259$, with conductor $45$ |
| Generic density of odd order reductions |
$299/2688$ |