| Curve name |
$X_{212g}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{212}$ |
| Curves that $X_{212g}$ minimally covers |
|
| Curves that minimally cover $X_{212g}$ |
|
| Curves that minimally cover $X_{212g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1855425871872t^{24} + 27831388078080t^{22} - 15713137852416t^{20} +
3913788948480t^{18} - 985242009600t^{16} + 149484994560t^{14} -
23130537984t^{12} + 2335703040t^{10} - 240537600t^{8} + 14929920t^{6} -
936576t^{4} + 25920t^{2} - 27\]
\[B(t) = 972777519512027136t^{36} + 30642491864628854784t^{34} -
63154540524569886720t^{32} + 30642491864628854784t^{30} -
10869648806891225088t^{28} + 2842809303847403520t^{26} -
632982247040876544t^{24} + 112216156730818560t^{22} - 18089358573699072t^{20} +
2348795207614464t^{18} - 282646227714048t^{16} + 27396522639360t^{14} -
2414635646976t^{12} + 169444638720t^{10} - 10123149312t^{8} + 445906944t^{6} -
14359680t^{4} + 108864t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 2751x - 104477$, with conductor $75$ |
| Generic density of odd order reductions |
$11/112$ |