| Curve name |
$X_{242g}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 20 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{242}$ |
| Curves that $X_{242g}$ minimally covers |
|
| Curves that minimally cover $X_{242g}$ |
|
| Curves that minimally cover $X_{242g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{32} - 5184t^{24} - 84672t^{16} - 497664t^{8} - 442368\]
\[B(t) = 432t^{48} + 31104t^{40} + 881280t^{32} + 12192768t^{24} +
80953344t^{16} + 191102976t^{8} - 113246208\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 12716x + 235824$, with conductor $18496$ |
| Generic density of odd order reductions |
$4769/28672$ |