| Curve name |
$X_{212a}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{212}$ |
| Curves that $X_{212a}$ minimally covers |
|
| Curves that minimally cover $X_{212a}$ |
|
| Curves that minimally cover $X_{212a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1855425871872t^{24} + 27831388078080t^{22} - 15597173735424t^{20} +
2174327193600t^{18} - 6794772480t^{16} - 40768634880t^{14} + 7842299904t^{12} -
637009920t^{10} - 1658880t^{8} + 8294400t^{6} - 929664t^{4} + 25920t^{2} - 27\]
\[B(t) = 972777519512027136t^{36} + 30642491864628854784t^{34} -
63245738417024139264t^{32} + 27769758252319899648t^{30} -
4944635731504005120t^{28} + 104735079615430656t^{26} + 108371714324299776t^{24}
- 24079716965154816t^{22} + 2644329759768576t^{20} - 41317652496384t^{16} +
5878837149696t^{14} - 413405282304t^{12} - 6242697216t^{10} + 4605050880t^{8} -
404103168t^{6} + 14380416t^{4} - 108864t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 990x + 22765$, with conductor $45$ |
| Generic density of odd order reductions |
$299/2688$ |