| Curve name |
$X_{229h}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{229}$ |
| Curves that $X_{229h}$ minimally covers |
|
| Curves that minimally cover $X_{229h}$ |
|
| Curves that minimally cover $X_{229h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27648t^{16} - 110592t^{14} + 1575936t^{12} + 3345408t^{10} -
4544640t^{8} + 836352t^{6} + 98496t^{4} - 1728t^{2} - 108\]
\[B(t) = -1769472t^{24} - 10616832t^{22} - 241532928t^{20} - 898007040t^{18} +
1176809472t^{16} + 3905667072t^{14} - 4125413376t^{12} + 976416768t^{10} +
73550592t^{8} - 14031360t^{6} - 943488t^{4} - 10368t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 14463x + 1157247$, with conductor $3264$ |
| Generic density of odd order reductions |
$271/2688$ |