| Curve name |
$X_{211o}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 13 & 13 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 15 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 11 & 11 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{211}$ |
| Curves that $X_{211o}$ minimally covers |
|
| Curves that minimally cover $X_{211o}$ |
|
| Curves that minimally cover $X_{211o}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{26} - 52704t^{25} - 1346112t^{24} - 12624768t^{23} -
59280768t^{22} - 161782272t^{21} - 280433664t^{20} - 210511872t^{19} +
1026763776t^{18} + 4221517824t^{17} + 4718297088t^{16} + 13078167552t^{15} -
3409772544t^{14} - 52312670208t^{13} + 75492753408t^{12} - 270177140736t^{11} +
262851526656t^{10} + 215564156928t^{9} - 1148656287744t^{8} + 2650640744448t^{7}
- 3885024411648t^{6} + 3309507182592t^{5} - 1411500736512t^{4} +
221056598016t^{3} - 1811939328t^{2}\]
\[B(t) = -432t^{39} + 430272t^{38} + 33965568t^{37} + 771828480t^{36} +
8831607552t^{35} + 59945287680t^{34} + 263218249728t^{33} + 804177248256t^{32} +
1708217745408t^{31} + 1631653134336t^{30} - 4031911821312t^{29} -
22897894883328t^{28} - 64224805257216t^{27} - 95590557941760t^{26} -
44811864244224t^{25} + 112684506808320t^{24} + 874052862345216t^{23} +
1037364029816832t^{22} + 4149456119267328t^{20} - 13984845797523456t^{19} +
7211808435732480t^{18} + 11471837246521344t^{17} - 97884731332362240t^{16} +
263064802333556736t^{15} - 375159109768445952t^{14} + 264235373121503232t^{13} +
427728079247376384t^{12} - 1791196130608939008t^{11} + 3372963849069133824t^{10}
- 4416069430828597248t^{9} + 4022860158357995520t^{8} - 2370716600434163712t^{7}
+ 828744519930347520t^{6} - 145881003750064128t^{5} + 7392016673538048t^{4} +
29686813949952t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 429480075x - 3503500199750$, with conductor $25200$ |
| Generic density of odd order reductions |
$139/1344$ |