| Curve name |
$X_{211n}$ |
| 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} 9 & 0 \\ 0 & 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} 15 & 0 \\ 24 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{211}$ |
| Curves that $X_{211n}$ minimally covers |
|
| Curves that minimally cover $X_{211n}$ |
|
| Curves that minimally cover $X_{211n}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{30} - 12960t^{29} - 230904t^{28} - 362880t^{27} + 11012976t^{26}
+ 49351680t^{25} - 118119168t^{24} - 1073295360t^{23} - 756843264t^{22} +
6877716480t^{21} + 11701352448t^{20} + 836075520t^{19} - 19700748288t^{18} -
188342599680t^{17} - 80146464768t^{16} + 753370398720t^{15} - 315211972608t^{14}
- 53508833280t^{13} + 2995546226688t^{12} - 7042781675520t^{11} -
3100030009344t^{10} + 17584871178240t^{9} - 7741057794048t^{8} -
12937246801920t^{7} + 11547942322176t^{6} + 1522029035520t^{5} -
3873926283264t^{4} + 869730877440t^{3} - 7247757312t^{2}\]
\[B(t) = 54t^{45} - 54432t^{44} - 3599640t^{43} - 46158336t^{42} -
14406336t^{41} + 3276370944t^{40} + 16190122752t^{39} - 61270401024t^{38} -
629420585472t^{37} - 376080703488t^{36} + 8834236876800t^{35} +
23400304607232t^{34} - 32051414237184t^{33} - 243210132652032t^{32} -
291096695144448t^{31} + 843056639115264t^{30} + 2602154330947584t^{29} +
1255153134993408t^{28} - 205235225100288t^{27} - 19225395609993216t^{26} -
71390109420748800t^{25} + 87010214380830720t^{24} + 285560437682995200t^{23} -
307606329759891456t^{22} + 13135054406418432t^{21} + 321319202558312448t^{20} -
2664606034890326016t^{19} + 3453159993816121344t^{18} +
4769328253246636032t^{17} - 15939019253483569152t^{16} +
8402085933792362496t^{15} + 24536997803832901632t^{14} -
37053475069309747200t^{13} - 6309587195850129408t^{12} +
42239700469240823808t^{11} - 16447148038180306944t^{10} -
17384011934516379648t^{9} + 14071906054044647424t^{8} + 247498967900749824t^{7}
- 3171976696924471296t^{6} + 989461508951900160t^{5} - 59848616923103232t^{4} -
237494511599616t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 847072809x - 9488980043537$, with conductor
$4410$ |
| Generic density of odd order reductions |
$271/2688$ |