| Curve name |
$X_{205k}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 7 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{205}$ |
| Curves that $X_{205k}$ minimally covers |
|
| Curves that minimally cover $X_{205k}$ |
|
| Curves that minimally cover $X_{205k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{24} + 6696t^{20} - 27540t^{16} + 41904t^{12} - 27540t^{8} +
6696t^{4} - 108\]
\[B(t) = 432t^{36} + 53136t^{32} - 611712t^{28} + 2274048t^{24} - 4139424t^{20}
+ 4139424t^{16} - 2274048t^{12} + 611712t^{8} - 53136t^{4} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 25072033x + 784578884063$, with conductor $196800$ |
| Generic density of odd order reductions |
$299/2688$ |