| Curve name |
$X_{211p}$ |
| Index |
$96$ |
| Level |
$16$ |
| 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 \\ 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 $X_{211p}$ minimally covers |
|
| Curves that minimally cover $X_{211p}$ |
|
| Curves that minimally cover $X_{211p}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{22} - 51840t^{21} - 928800t^{20} - 3939840t^{19} - 464832t^{18}
+ 39813120t^{17} + 69838848t^{16} - 26542080t^{15} - 105781248t^{14} -
982056960t^{13} - 853327872t^{12} + 3928227840t^{11} - 1692499968t^{10} +
1698693120t^{9} + 17878745088t^{8} - 40768634880t^{7} - 1903951872t^{6} +
64550338560t^{5} - 60869836800t^{4} + 13589544960t^{3} - 113246208t^{2}\]
\[B(t) = -432t^{33} + 435456t^{32} + 28766016t^{31} + 400619520t^{30} +
2187171072t^{29} + 1860268032t^{28} - 23140076544t^{27} - 80374726656t^{26} -
51186069504t^{25} + 366646984704t^{24} + 1194755899392t^{23} +
568085446656t^{22} - 1483392614400t^{21} - 6344363999232t^{20} -
27701586690048t^{19} + 19691250647040t^{18} + 110806346760192t^{17} -
101509823987712t^{16} + 94937127321600t^{15} + 145429874343936t^{14} -
1223430040977408t^{13} + 1501786049347584t^{12} + 838632562753536t^{11} -
5267438086127616t^{10} + 6066032225550336t^{9} + 1950632411922432t^{8} -
9173660375973888t^{7} + 6721280220856320t^{6} - 1930454655565824t^{5} +
116891829927936t^{4} + 463856467968t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 1106380812x + 14164624511984$, with conductor $20160$ |
| Generic density of odd order reductions |
$11/112$ |