| Curve name |
$X_{200c}$ |
| 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 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{200}$ |
| Curves that $X_{200c}$ minimally covers |
|
| Curves that minimally cover $X_{200c}$ |
|
| Curves that minimally cover $X_{200c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27648t^{16} + 110592t^{14} - 1741824t^{12} + 3290112t^{10} -
2885760t^{8} + 822528t^{6} - 108864t^{4} + 1728t^{2} - 108\]
\[B(t) = -1769472t^{24} + 10616832t^{22} + 204374016t^{20} - 885620736t^{18} +
2514530304t^{16} - 3459760128t^{14} + 2284498944t^{12} - 864940032t^{10} +
157158144t^{8} - 13837824t^{6} + 798336t^{4} + 10368t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 6657x - 71775$, with conductor $1344$ |
| Generic density of odd order reductions |
$271/2688$ |