| Curve name |
$X_{208k}$ |
| 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 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 14 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{208}$ |
| Curves that $X_{208k}$ minimally covers |
|
| Curves that minimally cover $X_{208k}$ |
|
| Curves that minimally cover $X_{208k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{24} + 216t^{20} - 216t^{12} + 216t^{4} - 108\]
\[B(t) = 432t^{36} - 1296t^{32} + 648t^{28} + 1512t^{24} - 2592t^{20} +
2592t^{16} - 1512t^{12} - 648t^{8} + 1296t^{4} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 1434672033x + 20915444484063$, with conductor
$196800$ |
| Generic density of odd order reductions |
$299/2688$ |