| Curve name |
$X_{76}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 6 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 6 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{23}$ |
| Curves that $X_{76}$ minimally covers |
$X_{23}$, $X_{29}$, $X_{49}$ |
| Curves that minimally cover $X_{76}$ |
$X_{257}$, $X_{258}$, $X_{76a}$, $X_{76b}$, $X_{76c}$, $X_{76d}$ |
| Curves that minimally cover $X_{76}$ and have infinitely many rational
points. |
$X_{76a}$, $X_{76b}$, $X_{76c}$, $X_{76d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{76}) = \mathbb{Q}(f_{76}), f_{23} =
\frac{2f_{76}^{2} + 8}{f_{76}^{2} + 4f_{76} - 4}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 333x - 9963$, with conductor $19200$ |
| Generic density of odd order reductions |
$401/1792$ |