| Curve name |
$X_{99d}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{99}$ |
| Curves that $X_{99d}$ minimally covers |
|
| Curves that minimally cover $X_{99d}$ |
|
| Curves that minimally cover $X_{99d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} - 1080t^{14} - 4536t^{12} - 10368t^{10} - 14040t^{8} -
11664t^{6} - 6156t^{4} - 2160t^{2} - 432\]
\[B(t) = -432t^{24} - 6480t^{22} - 43416t^{20} - 171288t^{18} - 440640t^{16} -
769824t^{14} - 914760t^{12} - 705672t^{10} - 296784t^{8} - 5616t^{6} +
59616t^{4} + 25920t^{2} + 3456\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 45075x - 3662750$, with conductor $3600$ |
| Generic density of odd order reductions |
$635/5376$ |