| Curve name |
$X_{203e}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 2 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 4 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{203}$ |
| Curves that $X_{203e}$ minimally covers |
|
| Curves that minimally cover $X_{203e}$ |
|
| Curves that minimally cover $X_{203e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -452984832t^{16} - 452984832t^{14} - 1783627776t^{12} - 842268672t^{10}
- 184688640t^{8} - 13160448t^{6} - 435456t^{4} - 1728t^{2} - 27\]
\[B(t) = -3710851743744t^{24} - 5566277615616t^{22} + 26787711025152t^{20} +
29020020277248t^{18} + 20599032250368t^{16} + 7085588742144t^{14} +
1169663459328t^{12} + 110712324096t^{10} + 5029060608t^{8} + 110702592t^{6} +
1596672t^{4} - 5184t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 27744x + 1787904$, with conductor $816$ |
| Generic density of odd order reductions |
$215/2688$ |