| Curve name |
$X_{12}$ |
| Index |
$6$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 2 & 3 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$3$ |
$X_{6}$ |
|
| Meaning/Special name |
Elliptic curves with discriminant $\Delta$ whose $2$-isogenous curve has
discriminant in the square class of $\Delta$ |
| Chosen covering |
$X_{6}$ |
| Curves that $X_{12}$ minimally covers |
$X_{6}$ |
| Curves that minimally cover $X_{12}$ |
$X_{25}$, $X_{30}$, $X_{31}$, $X_{40}$, $X_{51}$, $X_{52}$ |
| Curves that minimally cover $X_{12}$ and have infinitely many rational
points. |
$X_{25}$, $X_{30}$, $X_{31}$, $X_{40}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{12}) = \mathbb{Q}(f_{12}), f_{6} =
-f_{12}^{2} - 16\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 203x - 1048$, with conductor $2200$ |
| Generic density of odd order reductions |
$13/42$ |