| Curve name |
$X_{118j}$ |
| Index |
$48$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 16 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{118}$ |
| Curves that $X_{118j}$ minimally covers |
|
| Curves that minimally cover $X_{118j}$ |
|
| Curves that minimally cover $X_{118j}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{14} - 216t^{12} + 3456t^{8} + 6480t^{6} - 3456t^{4} - 6912t^{2}\]
\[B(t) = 54t^{21} + 648t^{19} + 1296t^{17} - 12096t^{15} - 55728t^{13} -
5184t^{11} + 314496t^{9} + 456192t^{7} + 165888t^{5} + 221184t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 - 167342x - 26305756$, with conductor $7605$ |
| Generic density of odd order reductions |
$25/224$ |