| Curve name |
$X_{99j}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 4 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{99}$ |
| Curves that $X_{99j}$ minimally covers |
|
| Curves that minimally cover $X_{99j}$ |
|
| Curves that minimally cover $X_{99j}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{12} - 864t^{10} - 2700t^{8} - 4104t^{6} - 3132t^{4} - 1296t^{2}
- 432\]
\[B(t) = -432t^{18} - 5184t^{16} - 26568t^{14} - 75600t^{12} - 128952t^{10} -
129600t^{8} - 63504t^{6} + 2592t^{4} + 15552t^{2} + 3456\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 1803x - 29302$, with conductor $720$ |
| Generic density of odd order reductions |
$635/5376$ |