| Curve name |
$X_{75f}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{75}$ |
| Curves that $X_{75f}$ minimally covers |
|
| Curves that minimally cover $X_{75f}$ |
|
| Curves that minimally cover $X_{75f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -442368t^{8} + 3317760t^{6} - 926208t^{4} + 51840t^{2} - 108\]
\[B(t) = -113246208t^{12} - 1783627776t^{10} + 1842020352t^{8} - 390168576t^{6}
+ 28781568t^{4} - 435456t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 55743x + 1756863$, with conductor $14784$ |
| Generic density of odd order reductions |
$307/2688$ |