| Curve name |
$X_{185a}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 8 & 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 $X_{185a}$ minimally covers |
|
| Curves that minimally cover $X_{185a}$ |
|
| Curves that minimally cover $X_{185a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -442368t^{24} - 6414336t^{20} - 414720t^{16} + 1022976t^{12} -
25920t^{8} - 25056t^{4} - 108\]
\[B(t) = 113246208t^{36} - 3652190208t^{32} - 4671406080t^{28} +
1734082560t^{24} + 554729472t^{20} - 138682368t^{16} - 27095040t^{12} +
4561920t^{8} + 222912t^{4} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 77772x + 8343664$, with conductor $2880$ |
| Generic density of odd order reductions |
$73/672$ |