| Curve name |
$X_{211a}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 11 & 11 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \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 $X_{211a}$ minimally covers |
|
| Curves that minimally cover $X_{211a}$ |
|
| Curves that minimally cover $X_{211a}$ 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 |
$y^2 + xy + y = x^3 - x^2 - 26842505x + 54748901247$, with conductor
$3150$ |
| Generic density of odd order reductions |
$11/112$ |