| Curve name |
$X_{61}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{24}$ |
| Curves that $X_{61}$ minimally covers |
$X_{24}$, $X_{35}$, $X_{45}$ |
| Curves that minimally cover $X_{61}$ |
$X_{209}$, $X_{210}$, $X_{251}$, $X_{254}$, $X_{392}$, $X_{393}$, $X_{61a}$, $X_{61b}$, $X_{61c}$, $X_{61d}$, $X_{61e}$, $X_{61f}$, $X_{61g}$, $X_{61h}$ |
| Curves that minimally cover $X_{61}$ and have infinitely many rational
points. |
$X_{209}$, $X_{210}$, $X_{61a}$, $X_{61b}$, $X_{61c}$, $X_{61d}$, $X_{61e}$, $X_{61f}$, $X_{61g}$, $X_{61h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{61}) = \mathbb{Q}(f_{61}), f_{24} =
f_{61}^{2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 4226x + 94223$, with conductor $7275$ |
| Generic density of odd order reductions |
$289/1792$ |