| Curve name |
$X_{211i}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 24 & 7 \end{matrix}\right],
\left[ \begin{matrix} 9 & 0 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 11 & 11 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{211}$ |
| Curves that $X_{211i}$ minimally covers |
|
| Curves that minimally cover $X_{211i}$ |
|
| Curves that minimally cover $X_{211i}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{26} - 50976t^{25} - 516672t^{24} + 2256768t^{23} +
13709952t^{22} + 24095232t^{21} - 107910144t^{20} - 287152128t^{19} -
1945949184t^{18} - 3212918784t^{17} + 1745584128t^{16} - 7345078272t^{15} +
54345793536t^{14} + 29380313088t^{13} + 27929346048t^{12} + 205626802176t^{11} -
498162991104t^{10} + 294043779072t^{9} - 441999949824t^{8} - 394776281088t^{7} +
898495414272t^{6} - 591598190592t^{5} - 541769859072t^{4} + 213808840704t^{3} -
1811939328t^{2}\]
\[B(t) = 432t^{39} - 440640t^{38} - 23514624t^{37} - 81665280t^{36} +
1006214400t^{35} + 7272529920t^{34} - 10946396160t^{33} - 74036477952t^{32} -
355781984256t^{31} - 695981113344t^{30} + 2242933161984t^{29} +
9949394239488t^{28} + 39489455259648t^{27} + 95714222800896t^{26} -
33410915500032t^{25} + 55437189709824t^{24} - 1232376548032512t^{23} -
2171796140851200t^{22} - 8687184563404800t^{20} + 19718024768520192t^{19} +
3547980141428736t^{18} + 8553194368008192t^{17} + 98011364148117504t^{16} -
161748808743518208t^{15} + 163010875219771392t^{14} - 146992867703783424t^{13} -
182447272976449536t^{12} + 373064449923219456t^{11} - 310531495619985408t^{10} +
183650052797890560t^{9} + 488051221337210880t^{8} - 270103621297766400t^{7} -
87687426704670720t^{6} + 100994541057736704t^{5} - 7570137557237760t^{4} -
29686813949952t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 6914880075x - 221322257999750$, with conductor $25200$ |
| Generic density of odd order reductions |
$139/1344$ |