| Curve name |
$X_{243c}$ |
| 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} 1 & 0 \\ 16 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 16 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{243}$ |
| Curves that $X_{243c}$ minimally covers |
|
| Curves that minimally cover $X_{243c}$ |
|
| Curves that minimally cover $X_{243c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} + 1728t^{8} - 1728\]
\[B(t) = 432t^{24} + 12960t^{16} - 41472t^{8} + 27648\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 5121x + 171615$, with conductor $16320$ |
| Generic density of odd order reductions |
$299/2688$ |