| Curve name |
$X_{211s}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 9 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 15 & 0 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{211}$ |
| Curves that $X_{211s}$ minimally covers |
|
| Curves that minimally cover $X_{211s}$ |
|
| Curves that minimally cover $X_{211s}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} - 12960t^{15} - 232416t^{14} - 1088640t^{13} - 1975104t^{12}
+ 1451520t^{11} + 5377536t^{10} + 22394880t^{9} + 48176640t^{8} - 89579520t^{7}
+ 86040576t^{6} - 92897280t^{5} - 505626624t^{4} + 1114767360t^{3} -
951975936t^{2} + 212336640t - 1769472\]
\[B(t) = -54t^{24} + 54432t^{23} + 3595104t^{22} + 50730624t^{21} +
316540224t^{20} + 838688256t^{19} + 733404672t^{18} - 2414168064t^{17} -
12561246720t^{16} - 20149420032t^{15} - 16335323136t^{14} - 1226244096t^{13} +
268429787136t^{12} + 4904976384t^{11} - 261365170176t^{10} + 1289562882048t^{9}
- 3215679160320t^{8} + 2472108097536t^{7} + 3004025536512t^{6} -
13741068386304t^{5} + 20744780120064t^{4} - 13298728697856t^{3} +
3769739771904t^{2} - 228304355328t - 905969664\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 30732800x + 65587209600$, with conductor $1680$ |
| Generic density of odd order reductions |
$19/336$ |