| Curve name |
$X_{78}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{33}$ |
| Curves that $X_{78}$ minimally covers |
$X_{33}$, $X_{36}$, $X_{44}$ |
| Curves that minimally cover $X_{78}$ |
$X_{201}$, $X_{202}$, $X_{233}$, $X_{234}$, $X_{331}$, $X_{332}$, $X_{78a}$, $X_{78b}$, $X_{78c}$, $X_{78d}$, $X_{78e}$, $X_{78f}$, $X_{78g}$, $X_{78h}$, $X_{78i}$, $X_{78j}$ |
| Curves that minimally cover $X_{78}$ and have infinitely many rational
points. |
$X_{202}$, $X_{233}$, $X_{234}$, $X_{78a}$, $X_{78b}$, $X_{78c}$, $X_{78d}$, $X_{78e}$, $X_{78f}$, $X_{78g}$, $X_{78h}$, $X_{78i}$, $X_{78j}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{78}) = \mathbb{Q}(f_{78}), f_{33} =
\frac{8f_{78}}{f_{78}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 425379x + 106683838$, with conductor $5544$ |
| Generic density of odd order reductions |
$643/5376$ |