| Curve name |
$X_{62}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{25}$ |
| Curves that $X_{62}$ minimally covers |
$X_{25}$, $X_{38}$, $X_{46}$ |
| Curves that minimally cover $X_{62}$ |
$X_{198}$, $X_{200}$, $X_{203}$, $X_{204}$, $X_{246}$, $X_{247}$, $X_{62a}$, $X_{62b}$, $X_{62c}$, $X_{62d}$, $X_{62e}$, $X_{62f}$, $X_{62g}$, $X_{62h}$, $X_{62i}$, $X_{62j}$ |
| Curves that minimally cover $X_{62}$ and have infinitely many rational
points. |
$X_{200}$, $X_{203}$, $X_{204}$, $X_{62a}$, $X_{62b}$, $X_{62c}$, $X_{62d}$, $X_{62e}$, $X_{62f}$, $X_{62g}$, $X_{62h}$, $X_{62i}$, $X_{62j}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{62}) = \mathbb{Q}(f_{62}), f_{25} =
\frac{f_{62}^{2} + 2}{f_{62}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 33339x + 754630$, with conductor $5544$ |
| Generic density of odd order reductions |
$643/5376$ |