| Curve name |
$X_{82}$ |
| 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 & 0 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{35}$ |
| Curves that $X_{82}$ minimally covers |
$X_{35}$, $X_{41}$, $X_{43}$ |
| Curves that minimally cover $X_{82}$ |
$X_{261}$, $X_{263}$, $X_{82a}$, $X_{82b}$, $X_{82c}$, $X_{82d}$ |
| Curves that minimally cover $X_{82}$ and have infinitely many rational
points. |
$X_{82a}$, $X_{82b}$, $X_{82c}$, $X_{82d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{82}) = \mathbb{Q}(f_{82}), f_{35} =
\frac{4f_{82}^{2} - 8}{f_{82}^{2} + 4f_{82} + 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 583x - 5213$, with conductor $19200$ |
| Generic density of odd order reductions |
$401/1792$ |