| Curve name |
$X_{229e}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{229}$ |
| Curves that $X_{229e}$ minimally covers |
|
| Curves that minimally cover $X_{229e}$ |
|
| Curves that minimally cover $X_{229e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -110592t^{24} - 1105920t^{22} + 2598912t^{20} + 46835712t^{18} +
121326336t^{16} + 30827520t^{14} - 131763456t^{12} + 7706880t^{10} +
7582896t^{8} + 731808t^{6} + 10152t^{4} - 1080t^{2} - 27\]
\[B(t) = -14155776t^{36} - 212336640t^{34} - 3089498112t^{32} -
27377270784t^{30} - 111635988480t^{28} - 144845438976t^{26} + 269705576448t^{24}
+ 823468032000t^{22} + 200518447104t^{20} - 595708342272t^{18} +
50129611776t^{16} + 51466752000t^{14} + 4214149632t^{12} - 565802496t^{10} -
109019520t^{8} - 6683904t^{6} - 188568t^{4} - 3240t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 1044928x + 708081972$, with conductor $13872$ |
| Generic density of odd order reductions |
$299/2688$ |