| Curve name |
$X_{235m}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 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} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{235}$ |
| Curves that $X_{235m}$ minimally covers |
|
| Curves that minimally cover $X_{235m}$ |
|
| Curves that minimally cover $X_{235m}$ 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 |
$y^2 + xy + y = x^3 - x^2 + 47245x - 2990253$, with conductor $3150$ |
| Generic density of odd order reductions |
$11/112$ |