| Curve name |
$X_{32d}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{32}$ |
| Curves that $X_{32d}$ minimally covers |
|
| Curves that minimally cover $X_{32d}$ |
|
| Curves that minimally cover $X_{32d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{8} - 5184t^{6} - 84672t^{4} - 497664t^{2} - 442368\]
\[B(t) = -432t^{12} - 31104t^{10} - 881280t^{8} - 12192768t^{6} - 80953344t^{4}
- 191102976t^{2} + 113246208\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 12008x - 502488$, with conductor $2400$ |
| Generic density of odd order reductions |
$289/1792$ |