| Curve name |
$X_{33c}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{33}$ |
| Curves that $X_{33c}$ minimally covers |
|
| Curves that minimally cover $X_{33c}$ |
|
| Curves that minimally cover $X_{33c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{8} - 648t^{6} - 5292t^{4} - 15552t^{2} - 6912\]
\[B(t) = 54t^{12} + 1944t^{10} + 27540t^{8} + 190512t^{6} + 632448t^{4} +
746496t^{2} - 221184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 156x + 736$, with conductor $288$ |
| Generic density of odd order reductions |
$643/5376$ |