| Curve name |
$X_{213g}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{213}$ |
| Curves that $X_{213g}$ minimally covers |
|
| Curves that minimally cover $X_{213g}$ |
|
| Curves that minimally cover $X_{213g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} - 864t^{14} - 1296t^{12} + 864t^{10} + 1080t^{8} + 864t^{6}
- 1296t^{4} - 864t^{2} - 108\]
\[B(t) = -432t^{24} - 5184t^{22} - 18144t^{20} - 12096t^{18} + 24624t^{16} +
31104t^{14} - 12096t^{12} + 31104t^{10} + 24624t^{8} - 12096t^{6} - 18144t^{4} -
5184t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 7262081x - 7534894881$, with conductor $16320$ |
| Generic density of odd order reductions |
$299/2688$ |