| Curve name |
$X_{205d}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 9 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{205}$ |
| Curves that $X_{205d}$ minimally covers |
|
| Curves that minimally cover $X_{205d}$ |
|
| Curves that minimally cover $X_{205d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{32} + 1620t^{28} - 3564t^{24} - 1620t^{20} + 7182t^{16} -
1620t^{12} - 3564t^{8} + 1620t^{4} - 27\]
\[B(t) = 54t^{48} + 6804t^{44} - 56376t^{40} + 74844t^{36} + 112590t^{32} -
258552t^{28} + 258552t^{20} - 112590t^{16} - 74844t^{12} + 56376t^{8} -
6804t^{4} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 + 198975145x - 1681901699853$, with conductor
$130050$ |
| Generic density of odd order reductions |
$299/2688$ |