| Curve name |
$X_{211r}$ |
| 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} 5 & 5 \\ 0 & 1 \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 $X_{211r}$ minimally covers |
|
| Curves that minimally cover $X_{211r}$ |
|
| Curves that minimally cover $X_{211r}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{22} - 12960t^{21} - 232200t^{20} - 984960t^{19} - 116208t^{18} +
9953280t^{17} + 17459712t^{16} - 6635520t^{15} - 26445312t^{14} -
245514240t^{13} - 213331968t^{12} + 982056960t^{11} - 423124992t^{10} +
424673280t^{9} + 4469686272t^{8} - 10192158720t^{7} - 475987968t^{6} +
16137584640t^{5} - 15217459200t^{4} + 3397386240t^{3} - 28311552t^{2}\]
\[B(t) = -54t^{33} + 54432t^{32} + 3595752t^{31} + 50077440t^{30} +
273396384t^{29} + 232533504t^{28} - 2892509568t^{27} - 10046840832t^{26} -
6398258688t^{25} + 45830873088t^{24} + 149344487424t^{23} + 71010680832t^{22} -
185424076800t^{21} - 793045499904t^{20} - 3462698336256t^{19} +
2461406330880t^{18} + 13850793345024t^{17} - 12688727998464t^{16} +
11867140915200t^{15} + 18178734292992t^{14} - 152928755122176t^{13} +
187723256168448t^{12} + 104829070344192t^{11} - 658429760765952t^{10} +
758254028193792t^{9} + 243829051490304t^{8} - 1146707546996736t^{7} +
840160027607040t^{6} - 241306831945728t^{5} + 14611478740992t^{4} +
57982058496t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 17287200x + 27669604050$, with conductor $630$ |
| Generic density of odd order reductions |
$271/2688$ |