| Curve name |
$X_{211e}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 11 & 11 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{211}$ |
| Curves that $X_{211e}$ minimally covers |
|
| Curves that minimally cover $X_{211e}$ |
|
| Curves that minimally cover $X_{211e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{26} - 13176t^{25} - 336528t^{24} - 3156192t^{23} - 14820192t^{22}
- 40445568t^{21} - 70108416t^{20} - 52627968t^{19} + 256690944t^{18} +
1055379456t^{17} + 1179574272t^{16} + 3269541888t^{15} - 852443136t^{14} -
13078167552t^{13} + 18873188352t^{12} - 67544285184t^{11} + 65712881664t^{10} +
53891039232t^{9} - 287164071936t^{8} + 662660186112t^{7} - 971256102912t^{6} +
827376795648t^{5} - 352875184128t^{4} + 55264149504t^{3} - 452984832t^{2}\]
\[B(t) = 54t^{39} - 53784t^{38} - 4245696t^{37} - 96478560t^{36} -
1103950944t^{35} - 7493160960t^{34} - 32902281216t^{33} - 100522156032t^{32} -
213527218176t^{31} - 203956641792t^{30} + 503988977664t^{29} +
2862236860416t^{28} + 8028100657152t^{27} + 11948819742720t^{26} +
5601483030528t^{25} - 14085563351040t^{24} - 109256607793152t^{23} -
129670503727104t^{22} - 518682014908416t^{20} + 1748105724690432t^{19} -
901476054466560t^{18} - 1433979655815168t^{17} + 12235591416545280t^{16} -
32883100291694592t^{15} + 46894888721055744t^{14} - 33029421640187904t^{13} -
53466009905922048t^{12} + 223899516326117376t^{11} - 421620481133641728t^{10} +
552008678853574656t^{9} - 502857519794749440t^{8} + 296339575054270464t^{7} -
103593064991293440t^{6} + 18235125468758016t^{5} - 924002084192256t^{4} -
3710851743744t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 - 2982500x - 2028731250$, with conductor $1050$ |
| Generic density of odd order reductions |
$1091/10752$ |