| Curve name |
$X_{213k}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \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 $X_{213k}$ minimally covers |
|
| Curves that minimally cover $X_{213k}$ |
|
| Curves that minimally cover $X_{213k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{24} - 540t^{22} - 3942t^{20} - 12204t^{18} - 12069t^{16} +
7560t^{14} + 14796t^{12} + 7560t^{10} - 12069t^{8} - 12204t^{6} - 3942t^{4} -
540t^{2} - 27\]
\[B(t) = 54t^{36} + 1620t^{34} + 19926t^{32} + 127872t^{30} + 445176t^{28} +
752976t^{26} + 234360t^{24} - 974592t^{22} - 1197180t^{20} - 589896t^{18} -
1197180t^{16} - 974592t^{14} + 234360t^{12} + 752976t^{10} + 445176t^{8} +
127872t^{6} + 19926t^{4} + 1620t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - 8220125x + 9061371057$, with conductor $50430$ |
| Generic density of odd order reductions |
$11/112$ |