| Curve name |
$X_{213i}$ |
| 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} 7 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{213}$ |
| Curves that $X_{213i}$ minimally covers |
|
| Curves that minimally cover $X_{213i}$ |
|
| Curves that minimally cover $X_{213i}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} - 216t^{14} - 324t^{12} + 216t^{10} + 270t^{8} + 216t^{6} -
324t^{4} - 216t^{2} - 27\]
\[B(t) = 54t^{24} + 648t^{22} + 2268t^{20} + 1512t^{18} - 3078t^{16} -
3888t^{14} + 1512t^{12} - 3888t^{10} - 3078t^{8} + 1512t^{6} + 2268t^{4} +
648t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 4890x + 129447$, with conductor $1230$ |
| Generic density of odd order reductions |
$19/336$ |