| Curve name |
$X_{223h}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{223}$ |
| Curves that $X_{223h}$ minimally covers |
|
| Curves that minimally cover $X_{223h}$ |
|
| Curves that minimally cover $X_{223h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -452984832t^{16} - 452984832t^{14} - 84934656t^{12} + 7077888t^{10} +
1105920t^{8} + 110592t^{6} - 20736t^{4} - 1728t^{2} - 27\]
\[B(t) = -3710851743744t^{24} - 5566277615616t^{22} - 2435246456832t^{20} -
202937204736t^{18} + 51640270848t^{16} + 8153726976t^{14} - 396361728t^{12} +
127401984t^{10} + 12607488t^{8} - 774144t^{6} - 145152t^{4} - 5184t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 544x - 4352$, with conductor $816$ |
| Generic density of odd order reductions |
$215/2688$ |