| Curve name |
$X_{205a}$ |
| 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 \\ 8 & 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 $X_{205a}$ minimally covers |
|
| Curves that minimally cover $X_{205a}$ |
|
| Curves that minimally cover $X_{205a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{32} + 6480t^{28} - 14256t^{24} - 6480t^{20} + 28728t^{16} -
6480t^{12} - 14256t^{8} + 6480t^{4} - 108\]
\[B(t) = 432t^{48} + 54432t^{44} - 451008t^{40} + 598752t^{36} + 900720t^{32} -
2068416t^{28} + 2068416t^{20} - 900720t^{16} - 598752t^{12} + 451008t^{8} -
54432t^{4} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 - 658532625x - 105614604984375$, with conductor
$252150$ |
| Generic density of odd order reductions |
$51/448$ |