| Curve name |
$X_{185b}$ |
| 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 & 6 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{185}$ |
| Curves that $X_{185b}$ minimally covers |
|
| Curves that minimally cover $X_{185b}$ |
|
| Curves that minimally cover $X_{185b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1769472t^{32} - 26542080t^{28} - 14598144t^{24} + 1658880t^{20} +
1838592t^{16} + 103680t^{12} - 57024t^{8} - 6480t^{4} - 27\]
\[B(t) = 905969664t^{48} - 28538044416t^{44} - 59114520576t^{40} -
19619905536t^{36} + 7378698240t^{32} + 4236115968t^{28} - 264757248t^{20} -
28823040t^{16} + 4790016t^{12} + 902016t^{8} + 27216t^{4} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 - 30380x + 2044622$, with conductor $225$ |
| Generic density of odd order reductions |
$299/2688$ |