| Curve name |
$X_{235s}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 0 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 15 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 0 & 15 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{235}$ |
| Curves that $X_{235s}$ minimally covers |
|
| Curves that minimally cover $X_{235s}$ |
|
| Curves that minimally cover $X_{235s}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{26} + 432t^{25} + 432t^{24} - 3024t^{23} + 1512t^{22} +
1296t^{21} - 1296t^{20} + 11664t^{19} - 6804t^{18} - 2592t^{17} - 6048t^{16} -
9504t^{15} - 3024t^{14} + 9504t^{13} - 6048t^{12} + 2592t^{11} - 6804t^{10} -
11664t^{9} - 1296t^{8} - 1296t^{7} + 1512t^{6} + 3024t^{5} + 432t^{4} - 432t^{3}
- 108t^{2}\]
\[B(t) = 432t^{39} - 2592t^{38} + 25056t^{36} - 34992t^{35} - 41472t^{34} +
96768t^{33} - 124416t^{32} + 202176t^{31} + 155520t^{30} - 414720t^{29} +
155520t^{28} - 451008t^{27} - 290304t^{26} + 870912t^{25} - 373248t^{24} +
1246752t^{23} - 388800t^{22} - 388800t^{20} - 1246752t^{19} - 373248t^{18} -
870912t^{17} - 290304t^{16} + 451008t^{15} + 155520t^{14} + 414720t^{13} +
155520t^{12} - 202176t^{11} - 124416t^{10} - 96768t^{9} - 41472t^{8} +
34992t^{7} + 25056t^{6} - 2592t^{4} - 432t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + 257224x - 37815802$, with conductor $7350$ |
| Generic density of odd order reductions |
$1091/10752$ |