| Curve name |
$X_{211j}$ |
| 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} 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 $X_{211j}$ minimally covers |
|
| Curves that minimally cover $X_{211j}$ |
|
| Curves that minimally cover $X_{211j}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} - 51840t^{15} - 929664t^{14} - 4354560t^{13} - 7900416t^{12}
+ 5806080t^{11} + 21510144t^{10} + 89579520t^{9} + 192706560t^{8} -
358318080t^{7} + 344162304t^{6} - 371589120t^{5} - 2022506496t^{4} +
4459069440t^{3} - 3807903744t^{2} + 849346560t - 7077888\]
\[B(t) = 432t^{24} - 435456t^{23} - 28760832t^{22} - 405844992t^{21} -
2532321792t^{20} - 6709506048t^{19} - 5867237376t^{18} + 19313344512t^{17} +
100489973760t^{16} + 161195360256t^{15} + 130682585088t^{14} + 9809952768t^{13}
- 2147438297088t^{12} - 39239811072t^{11} + 2090921361408t^{10} -
10316503056384t^{9} + 25725433282560t^{8} - 19776864780288t^{7} -
24032204292096t^{6} + 109928547090432t^{5} - 165958240960512t^{4} +
106389829582848t^{3} - 30157918175232t^{2} + 1826434842624t + 7247757312\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 122931201x - 524574745599$, with conductor $6720$ |
| Generic density of odd order reductions |
$299/2688$ |