| Curve name |
$X_{205b}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 9 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{205}$ |
| Curves that $X_{205b}$ minimally covers |
|
| Curves that minimally cover $X_{205b}$ |
|
| Curves that minimally cover $X_{205b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{24} + 6264t^{20} - 1620t^{16} - 15984t^{12} - 1620t^{8} +
6264t^{4} - 108\]
\[B(t) = 432t^{36} + 55728t^{32} - 285120t^{28} - 423360t^{24} + 541728t^{20} +
541728t^{16} - 423360t^{12} - 285120t^{8} + 55728t^{4} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 + 56597375x + 255124305025$, with conductor $277440$ |
| Generic density of odd order reductions |
$5/42$ |