| Curve name |
$X_{243p}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 16 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{243}$ |
| Curves that $X_{243p}$ minimally covers |
|
| Curves that minimally cover $X_{243p}$ |
|
| Curves that minimally cover $X_{243p}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{24} - 216t^{20} + 1620t^{16} + 3456t^{12} - 3456t^{4} - 1728\]
\[B(t) = 432t^{36} + 1296t^{32} + 14256t^{28} + 39312t^{24} - 2592t^{20} -
111456t^{16} - 96768t^{12} + 41472t^{8} + 82944t^{4} + 27648\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 1480065x + 852024705$, with conductor $277440$ |
| Generic density of odd order reductions |
$5/42$ |