| Curve name |
$X_{79h}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 4 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{79}$ |
| Curves that $X_{79h}$ minimally covers |
|
| Curves that minimally cover $X_{79h}$ |
|
| Curves that minimally cover $X_{79h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -891t^{8} - 3888t^{7} - 6264t^{6} - 2592t^{5} + 2808t^{4} - 5184t^{3} -
25056t^{2} - 31104t - 14256\]
\[B(t) = 10206t^{12} + 66096t^{11} + 169128t^{10} + 135648t^{9} - 382968t^{8} -
1503360t^{7} - 2721600t^{6} - 3006720t^{5} - 1531872t^{4} + 1085184t^{3} +
2706048t^{2} + 2115072t + 653184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 13067x + 574926$, with conductor $1120$ |
| Generic density of odd order reductions |
$307/2688$ |