| Curve name |
$X_{96}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{25}$ |
| Curves that $X_{96}$ minimally covers |
$X_{25}$, $X_{32}$, $X_{36}$ |
| Curves that minimally cover $X_{96}$ |
$X_{184}$, $X_{185}$, $X_{192}$, $X_{193}$, $X_{194}$, $X_{206}$, $X_{208}$, $X_{215}$, $X_{246}$, $X_{249}$, $X_{313}$, $X_{314}$, $X_{315}$, $X_{316}$, $X_{386}$, $X_{404}$, $X_{96a}$, $X_{96b}$, $X_{96c}$, $X_{96d}$, $X_{96e}$, $X_{96f}$, $X_{96g}$, $X_{96h}$, $X_{96i}$, $X_{96j}$, $X_{96k}$, $X_{96l}$, $X_{96m}$, $X_{96n}$, $X_{96o}$, $X_{96p}$, $X_{96q}$, $X_{96r}$, $X_{96s}$, $X_{96t}$ |
| Curves that minimally cover $X_{96}$ and have infinitely many rational
points. |
$X_{185}$, $X_{192}$, $X_{193}$, $X_{194}$, $X_{208}$, $X_{215}$, $X_{96a}$, $X_{96b}$, $X_{96c}$, $X_{96d}$, $X_{96e}$, $X_{96f}$, $X_{96g}$, $X_{96h}$, $X_{96i}$, $X_{96j}$, $X_{96k}$, $X_{96l}$, $X_{96m}$, $X_{96n}$, $X_{96o}$, $X_{96p}$, $X_{96q}$, $X_{96r}$, $X_{96s}$, $X_{96t}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{96}) = \mathbb{Q}(f_{96}), f_{25} =
-f_{96}^{2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 40148415x - 97904904944$, with conductor $6435$ |
| Generic density of odd order reductions |
$19/168$ |