| Curve name |
$X_{211f}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 24 & 7 \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} 9 & 0 \\ 16 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{211}$ |
| Curves that $X_{211f}$ minimally covers |
|
| Curves that minimally cover $X_{211f}$ |
|
| Curves that minimally cover $X_{211f}$ 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 |
$y^2 + xy + y = x^3 - x^2 - 432180005x + 3458268326247$, with conductor
$3150$ |
| Generic density of odd order reductions |
$11/112$ |