| Curve name |
$X_{235q}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 15 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \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 $X_{235q}$ minimally covers |
|
| Curves that minimally cover $X_{235q}$ |
|
| Curves that minimally cover $X_{235q}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{28} + 108t^{27} + 162t^{26} - 972t^{25} + 135t^{24} + 1944t^{23}
- 972t^{22} + 1512t^{21} - 675t^{20} - 6156t^{19} + 1566t^{18} + 1836t^{17} +
567t^{16} + 6480t^{15} - 1512t^{14} - 6480t^{13} + 567t^{12} - 1836t^{11} +
1566t^{10} + 6156t^{9} - 675t^{8} - 1512t^{7} - 972t^{6} - 1944t^{5} + 135t^{4}
+ 972t^{3} + 162t^{2} - 108t - 27\]
\[B(t) = -54t^{42} + 324t^{41} + 162t^{40} - 4104t^{39} + 4212t^{38} +
15552t^{37} - 25164t^{36} - 9720t^{35} + 24138t^{34} - 47412t^{33} + 86994t^{32}
+ 80352t^{31} - 162864t^{30} + 20736t^{29} - 97200t^{28} - 101088t^{27} +
288036t^{26} + 36936t^{25} + 84564t^{24} + 6480t^{23} - 202824t^{22} -
202824t^{20} - 6480t^{19} + 84564t^{18} - 36936t^{17} + 288036t^{16} +
101088t^{15} - 97200t^{14} - 20736t^{13} - 162864t^{12} - 80352t^{11} +
86994t^{10} + 47412t^{9} + 24138t^{8} + 9720t^{7} - 25164t^{6} - 15552t^{5} +
4212t^{4} + 4104t^{3} + 162t^{2} - 324t - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 755925x + 190620250$, with conductor $25200$ |
| Generic density of odd order reductions |
$139/1344$ |