| Curve name |
$X_{87a}$ |
| 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} 7 & 6 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 2 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{87}$ |
| Curves that $X_{87a}$ minimally covers |
|
| Curves that minimally cover $X_{87a}$ |
|
| Curves that minimally cover $X_{87a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -6912t^{16} - 34560t^{14} - 72576t^{12} - 82944t^{10} - 56160t^{8} -
23328t^{6} - 6156t^{4} - 1080t^{2} - 108\]
\[B(t) = 221184t^{24} + 1658880t^{22} + 5557248t^{20} + 10962432t^{18} +
14100480t^{16} + 12317184t^{14} + 7318080t^{12} + 2822688t^{10} + 593568t^{8} +
5616t^{6} - 29808t^{4} - 6480t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 4575x + 22750$, with conductor $3600$ |
| Generic density of odd order reductions |
$635/5376$ |