| Curve name |
$X_{33a}$ |
| 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 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{33}$ |
| Curves that $X_{33a}$ minimally covers |
|
| Curves that minimally cover $X_{33a}$ |
|
| Curves that minimally cover $X_{33a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{8} - 2592t^{6} - 21168t^{4} - 62208t^{2} - 27648\]
\[B(t) = 432t^{12} + 15552t^{10} + 220320t^{8} + 1524096t^{6} + 5059584t^{4} +
5971968t^{2} - 1769472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 1164x + 15280$, with conductor $576$ |
| Generic density of odd order reductions |
$289/1792$ |