| Curve name |
$X_{75e}$ |
| 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} 5 & 5 \\ 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 $X_{75e}$ minimally covers |
|
| Curves that minimally cover $X_{75e}$ |
|
| Curves that minimally cover $X_{75e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7077888t^{12} + 54853632t^{10} - 28200960t^{8} + 5363712t^{6} -
440640t^{4} + 13392t^{2} - 27\]
\[B(t) = 7247757312t^{18} + 111434268672t^{16} - 160356630528t^{14} +
74516004864t^{12} - 16955080704t^{10} + 2119385088t^{8} - 145539072t^{6} +
4893696t^{4} - 53136t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 + 682848x + 74301228$, with conductor $12936$ |
| Generic density of odd order reductions |
$691/5376$ |