| Curve name |
$X_{76b}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{76}$ |
| Curves that $X_{76b}$ minimally covers |
|
| Curves that minimally cover $X_{76b}$ |
|
| Curves that minimally cover $X_{76b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -270t^{12} + 3888t^{11} + 52272t^{10} + 129600t^{9} - 64800t^{8} -
870912t^{7} - 2225664t^{6} + 3483648t^{5} - 1036800t^{4} - 8294400t^{3} +
13381632t^{2} - 3981312t - 1105920\]
\[B(t) = -7344t^{18} - 93312t^{17} - 554688t^{16} - 1244160t^{15} +
12856320t^{14} + 90574848t^{13} + 98316288t^{12} - 258785280t^{11} -
80289792t^{10} - 321159168t^{8} + 4140564480t^{7} + 6292242432t^{6} -
23187161088t^{5} + 13164871680t^{4} + 5096079360t^{3} - 9088008192t^{2} +
6115295232t - 1925185536\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 13x + 85$, with conductor $768$ |
| Generic density of odd order reductions |
$417/1792$ |