| Curve name |
$X_{235h}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 9 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{235}$ |
| Curves that $X_{235h}$ minimally covers |
|
| Curves that minimally cover $X_{235h}$ |
|
| Curves that minimally cover $X_{235h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{22} + 1080t^{20} - 3132t^{18} + 2592t^{16} + 1512t^{14} -
3888t^{12} + 1512t^{10} + 2592t^{8} - 3132t^{6} + 1080t^{4} - 108t^{2}\]
\[B(t) = 432t^{33} - 6480t^{31} + 34992t^{29} - 82512t^{27} + 71280t^{25} +
50544t^{23} - 142992t^{21} + 112752t^{19} - 112752t^{17} + 142992t^{15} -
50544t^{13} - 71280t^{11} + 82512t^{9} - 34992t^{7} + 6480t^{5} - 432t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 1890x - 24300$, with conductor $630$ |
| Generic density of odd order reductions |
$271/2688$ |