| Curve name |
$X_{8d}$ |
| Index |
$12$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 2 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$6$ |
$X_{8}$ |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{8}$ |
| Curves that $X_{8d}$ minimally covers |
|
| Curves that minimally cover $X_{8d}$ |
|
| Curves that minimally cover $X_{8d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{2} - 324\]
\[B(t) = 432t^{3} - 3888t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 - 11x$, with conductor $33$ |
| Generic density of odd order reductions |
$5/42$ |