| Curve name |
$X_{77}$ |
| 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} 1 & 3 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{29}$ |
| Curves that $X_{77}$ minimally covers |
$X_{29}$, $X_{43}$, $X_{45}$ |
| Curves that minimally cover $X_{77}$ |
$X_{216}$, $X_{257}$, $X_{258}$, $X_{275}$, $X_{276}$, $X_{373}$, $X_{391}$ |
| Curves that minimally cover $X_{77}$ and have infinitely many rational
points. |
$X_{216}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{77}) = \mathbb{Q}(f_{77}), f_{29} =
\frac{f_{77}^{2} - \frac{1}{2}}{f_{77}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 + 307x - 4587$, with conductor $5376$ |
| Generic density of odd order reductions |
$401/1792$ |