| Curve name |
$X_{235t}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 15 \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 $X_{235t}$ minimally covers |
|
| Curves that minimally cover $X_{235t}$ |
|
| Curves that minimally cover $X_{235t}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{26} + 108t^{25} + 108t^{24} - 756t^{23} + 378t^{22} + 324t^{21} -
324t^{20} + 2916t^{19} - 1701t^{18} - 648t^{17} - 1512t^{16} - 2376t^{15} -
756t^{14} + 2376t^{13} - 1512t^{12} + 648t^{11} - 1701t^{10} - 2916t^{9} -
324t^{8} - 324t^{7} + 378t^{6} + 756t^{5} + 108t^{4} - 108t^{3} - 27t^{2}\]
\[B(t) = 54t^{39} - 324t^{38} + 3132t^{36} - 4374t^{35} - 5184t^{34} +
12096t^{33} - 15552t^{32} + 25272t^{31} + 19440t^{30} - 51840t^{29} +
19440t^{28} - 56376t^{27} - 36288t^{26} + 108864t^{25} - 46656t^{24} +
155844t^{23} - 48600t^{22} - 48600t^{20} - 155844t^{19} - 46656t^{18} -
108864t^{17} - 36288t^{16} + 56376t^{15} + 19440t^{14} + 51840t^{13} +
19440t^{12} - 25272t^{11} - 15552t^{10} - 12096t^{9} - 5184t^{8} + 4374t^{7} +
3132t^{6} - 324t^{4} - 54t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 54252075x + 153763452250$, with conductor $25200$ |
| Generic density of odd order reductions |
$139/1344$ |