| Curve name |
$X_{87h}$ |
| 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 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{87}$ |
| Curves that $X_{87h}$ minimally covers |
|
| Curves that minimally cover $X_{87h}$ |
|
| Curves that minimally cover $X_{87h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1728t^{16} - 8640t^{14} - 18144t^{12} - 20736t^{10} - 14040t^{8} -
5832t^{6} - 1539t^{4} - 270t^{2} - 27\]
\[B(t) = -27648t^{24} - 207360t^{22} - 694656t^{20} - 1370304t^{18} -
1762560t^{16} - 1539648t^{14} - 914760t^{12} - 352836t^{10} - 74196t^{8} -
702t^{6} + 3726t^{4} + 810t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 18300x - 182000$, with conductor $14400$ |
| Generic density of odd order reductions |
$41/336$ |