| Curve name |
$X_{211d}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 9 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 24 & 7 \end{matrix}\right],
\left[ \begin{matrix} 11 & 11 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 15 & 0 \\ 24 & 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 $X_{211d}$ minimally covers |
|
| Curves that minimally cover $X_{211d}$ |
|
| Curves that minimally cover $X_{211d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{24} - 51840t^{23} - 924480t^{22} - 1866240t^{21} +
36657792t^{20} + 183306240t^{19} - 164422656t^{18} - 2796871680t^{17} -
4929278976t^{16} + 2202992640t^{15} + 10001940480t^{14} + 65718190080t^{13} +
80080994304t^{12} - 262872760320t^{11} + 160031047680t^{10} - 140991528960t^{9}
- 1261895417856t^{8} + 2863996600320t^{7} - 673475198976t^{6} -
3003289436160t^{5} + 2402405056512t^{4} + 489223618560t^{3} - 969387540480t^{2}
+ 217432719360t - 1811939328\]
\[B(t) = 432t^{36} - 435456t^{35} - 28791936t^{34} - 374492160t^{33} -
460774656t^{32} + 21737963520t^{31} + 125373726720t^{30} - 211359891456t^{29} -
3510605463552t^{28} - 6612354072576t^{27} + 22499200991232t^{26} +
119390692442112t^{25} + 190112078757888t^{24} - 209126789480448t^{23} -
1352069013307392t^{22} - 1919012258709504t^{21} - 3093024429047808t^{20} +
4692168186789888t^{19} + 38308310729883648t^{18} - 18768672747159552t^{17} -
49488390864764928t^{16} + 122816784557408256t^{15} - 346129667406692352t^{14} +
214145832427978752t^{13} + 778699074592309248t^{12} - 1956097104971563008t^{11}
+ 1474507636161380352t^{10} + 1733388946001362944t^{9} -
3681136634549501952t^{8} + 886507638173466624t^{7} + 2103422093906411520t^{6} -
1458810037500641280t^{5} - 123688254896603136t^{4} + 402107894952099840t^{3} -
123660423508525056t^{2} + 7481077115387904t + 29686813949952\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 374124865x + 2848350713375$, with conductor $47040$ |
| Generic density of odd order reductions |
$271/2688$ |