| Curve name |
$X_{200h}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{200}$ |
| Curves that $X_{200h}$ minimally covers |
|
| Curves that minimally cover $X_{200h}$ |
|
| Curves that minimally cover $X_{200h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -442368t^{24} + 4423680t^{22} - 42688512t^{20} + 237330432t^{18} -
629462016t^{16} + 832204800t^{14} - 600044544t^{12} + 208051200t^{10} -
39341376t^{8} + 3708288t^{6} - 166752t^{4} + 4320t^{2} - 108\]
\[B(t) = 113246208t^{36} - 1698693120t^{34} - 3822059520t^{32} +
151976411136t^{30} - 1011826556928t^{28} + 3649896972288t^{26} -
8480605077504t^{24} + 12985617088512t^{22} - 12688948297728t^{20} +
7860634288128t^{18} - 3172237074432t^{16} + 811601068032t^{14} -
132509454336t^{12} + 14257410048t^{10} - 988111872t^{8} + 37103616t^{6} -
233280t^{4} - 25920t^{2} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 326209x - 25271231$, with conductor $9408$ |
| Generic density of odd order reductions |
$109/896$ |