| Curve name |
$X_{25}$ |
| Index |
$12$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$6$ |
$X_{8}$ |
|
| Meaning/Special name |
$\tilde{X}_{1}(2,4)$ |
| Chosen covering |
$X_{8}$ |
| Curves that $X_{25}$ minimally covers |
$X_{8}$, $X_{12}$, $X_{13}$ |
| Curves that minimally cover $X_{25}$ |
$X_{58}$, $X_{62}$, $X_{87}$, $X_{96}$, $X_{98}$, $X_{99}$, $X_{100}$, $X_{101}$, $X_{129}$, $X_{142}$, $X_{25a}$, $X_{25b}$, $X_{25c}$, $X_{25d}$, $X_{25e}$, $X_{25f}$, $X_{25g}$, $X_{25h}$, $X_{25i}$, $X_{25j}$, $X_{25k}$, $X_{25l}$, $X_{25m}$, $X_{25n}$ |
| Curves that minimally cover $X_{25}$ and have infinitely many rational
points. |
$X_{58}$, $X_{62}$, $X_{87}$, $X_{96}$, $X_{98}$, $X_{99}$, $X_{100}$, $X_{101}$, $X_{25a}$, $X_{25b}$, $X_{25c}$, $X_{25d}$, $X_{25e}$, $X_{25f}$, $X_{25g}$, $X_{25h}$, $X_{25i}$, $X_{25j}$, $X_{25k}$, $X_{25l}$, $X_{25m}$, $X_{25n}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{25}) = \mathbb{Q}(f_{25}), f_{8} =
-2f_{25}^{2} + 1\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 2430x - 42449$, with conductor $3465$ |
| Generic density of odd order reductions |
$5/42$ |