| Curve name |
$X_{233b}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{233}$ |
| Curves that $X_{233b}$ minimally covers |
|
| Curves that minimally cover $X_{233b}$ |
|
| Curves that minimally cover $X_{233b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -452984832t^{16} - 14042529792t^{14} - 25565331456t^{12} -
14219476992t^{10} - 2838896640t^{8} - 222179328t^{6} - 6241536t^{4} - 53568t^{2}
- 27\]
\[B(t) = -3710851743744t^{24} + 228217382240256t^{22} + 1254151925268480t^{20} +
1946776605032448t^{18} + 1331979249254400t^{16} + 444630885728256t^{14} +
76481562673152t^{12} + 6947357589504t^{10} + 325190246400t^{8} + 7426363392t^{6}
+ 74753280t^{4} + 212544t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 443904x + 113984640$, with conductor $816$ |
| Generic density of odd order reductions |
$215/2688$ |