| Curve name |
$X_{231a}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 4 & 13 \end{matrix}\right],
\left[ \begin{matrix} 3 & 9 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 12 & 15 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{231}$ |
| Curves that $X_{231a}$ minimally covers |
|
| Curves that minimally cover $X_{231a}$ |
|
| Curves that minimally cover $X_{231a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 81t^{16} + 1296t^{15} + 3456t^{14} - 42336t^{13} - 323568t^{12} -
737856t^{11} + 483840t^{10} + 5864832t^{9} + 12559968t^{8} + 11729664t^{7} +
1935360t^{6} - 5902848t^{5} - 5177088t^{4} - 1354752t^{3} + 221184t^{2} +
165888t + 20736\]
\[B(t) = 3888t^{23} + 89424t^{22} + 766368t^{21} + 2322432t^{20} - 6640704t^{19}
- 71475264t^{18} - 165214080t^{17} + 315850752t^{16} + 2844495360t^{15} +
7593896448t^{14} + 9354175488t^{13} - 18708350976t^{11} - 30375585792t^{10} -
22755962880t^{9} - 5053612032t^{8} + 5286850560t^{7} + 4574416896t^{6} +
850010112t^{5} - 594542592t^{4} - 392380416t^{3} - 91570176t^{2} - 7962624t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 240097x - 65021152$, with conductor $168640$ |
| Generic density of odd order reductions |
$13411/86016$ |