| Curve name |
$X_{213e}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{213}$ |
| Curves that $X_{213e}$ minimally covers |
|
| Curves that minimally cover $X_{213e}$ |
|
| Curves that minimally cover $X_{213e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{24} - 2160t^{22} - 15768t^{20} - 48816t^{18} - 48276t^{16} +
30240t^{14} + 59184t^{12} + 30240t^{10} - 48276t^{8} - 48816t^{6} - 15768t^{4} -
2160t^{2} - 108\]
\[B(t) = -432t^{36} - 12960t^{34} - 159408t^{32} - 1022976t^{30} - 3561408t^{28}
- 6023808t^{26} - 1874880t^{24} + 7796736t^{22} + 9577440t^{20} + 4719168t^{18}
+ 9577440t^{16} + 7796736t^{14} - 1874880t^{12} - 6023808t^{10} - 3561408t^{8} -
1022976t^{6} - 159408t^{4} - 12960t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - 32792836x - 72282118834$, with conductor $8670$ |
| Generic density of odd order reductions |
$271/2688$ |