| Curve name |
$X_{3}$ |
| Index |
$2$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 0 & 1 \\ 3 & 1 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$1$ |
$X_{1}$ |
|
| Meaning/Special name |
Elliptic curves whose discriminant is minus a square |
| Chosen covering |
$X_{1}$ |
| Curves that $X_{3}$ minimally covers |
$X_{1}$ |
| Curves that minimally cover $X_{3}$ |
$X_{10}$, $X_{20}$, $X_{3a}$, $X_{3b}$ |
| Curves that minimally cover $X_{3}$ and have infinitely many rational
points. |
$X_{10}$, $X_{20}$, $X_{3a}$, $X_{3b}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{3}) = \mathbb{Q}(f_{3}), f_{1} = -f_{3}^{2}
+ 1728\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 30x - 76$, with conductor $121$ |
| Generic density of odd order reductions |
$59/112$ |