| Curve name |
$X_{211b}$ |
| 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} 11 & 11 \\ 8 & 5 \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 $X_{211b}$ minimally covers |
|
| Curves that minimally cover $X_{211b}$ |
|
| Curves that minimally cover $X_{211b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{26} - 12744t^{25} - 129168t^{24} + 564192t^{23} + 3427488t^{22} +
6023808t^{21} - 26977536t^{20} - 71788032t^{19} - 486487296t^{18} -
803229696t^{17} + 436396032t^{16} - 1836269568t^{15} + 13586448384t^{14} +
7345078272t^{13} + 6982336512t^{12} + 51406700544t^{11} - 124540747776t^{10} +
73510944768t^{9} - 110499987456t^{8} - 98694070272t^{7} + 224623853568t^{6} -
147899547648t^{5} - 135442464768t^{4} + 53452210176t^{3} - 452984832t^{2}\]
\[B(t) = -54t^{39} + 55080t^{38} + 2939328t^{37} + 10208160t^{36} -
125776800t^{35} - 909066240t^{34} + 1368299520t^{33} + 9254559744t^{32} +
44472748032t^{31} + 86997639168t^{30} - 280366645248t^{29} - 1243674279936t^{28}
- 4936181907456t^{27} - 11964277850112t^{26} + 4176364437504t^{25} -
6929648713728t^{24} + 154047068504064t^{23} + 271474517606400t^{22} +
1085898070425600t^{20} - 2464753096065024t^{19} - 443497517678592t^{18} -
1069149296001024t^{17} - 12251420518514688t^{16} + 20218601092939776t^{15} -
20376359402471424t^{14} + 18374108462972928t^{13} + 22805909122056192t^{12} -
46633056240402432t^{11} + 38816436952498176t^{10} - 22956256599736320t^{9} -
61006402667151360t^{8} + 33762952662220800t^{7} + 10960928338083840t^{6} -
12624317632217088t^{5} + 946267194654720t^{4} + 3710851743744t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 - 48020000x - 128100018750$, with conductor
$1050$ |
| Generic density of odd order reductions |
$1091/10752$ |