| Curve name |
$X_{211h}$ |
| 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} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 11 & 11 \\ 8 & 5 \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 $X_{211h}$ minimally covers |
|
| Curves that minimally cover $X_{211h}$ |
|
| Curves that minimally cover $X_{211h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{24} - 12960t^{23} - 231120t^{22} - 466560t^{21} + 9164448t^{20} +
45826560t^{19} - 41105664t^{18} - 699217920t^{17} - 1232319744t^{16} +
550748160t^{15} + 2500485120t^{14} + 16429547520t^{13} + 20020248576t^{12} -
65718190080t^{11} + 40007761920t^{10} - 35247882240t^{9} - 315473854464t^{8} +
715999150080t^{7} - 168368799744t^{6} - 750822359040t^{5} + 600601264128t^{4} +
122305904640t^{3} - 242346885120t^{2} + 54358179840t - 452984832\]
\[B(t) = 54t^{36} - 54432t^{35} - 3598992t^{34} - 46811520t^{33} -
57596832t^{32} + 2717245440t^{31} + 15671715840t^{30} - 26419986432t^{29} -
438825682944t^{28} - 826544259072t^{27} + 2812400123904t^{26} +
14923836555264t^{25} + 23764009844736t^{24} - 26140848685056t^{23} -
169008626663424t^{22} - 239876532338688t^{21} - 386628053630976t^{20} +
586521023348736t^{19} + 4788538841235456t^{18} - 2346084093394944t^{17} -
6186048858095616t^{16} + 15352098069676032t^{15} - 43266208425836544t^{14} +
26768229053497344t^{13} + 97337384324038656t^{12} - 244512138121445376t^{11} +
184313454520172544t^{10} + 216673618250170368t^{9} - 460142079318687744t^{8} +
110813454771683328t^{7} + 262927761738301440t^{6} - 182351254687580160t^{5} -
15461031862075392t^{4} + 50263486869012480t^{3} - 15457552938565632t^{2} +
935134639423488t + 3710851743744\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 94119201x + 351412332249$, with conductor
$1470$ |
| Generic density of odd order reductions |
$11/112$ |