| Curve name |
$X_{85d}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{85}$ |
| Curves that $X_{85d}$ minimally covers |
|
| Curves that minimally cover $X_{85d}$ |
|
| Curves that minimally cover $X_{85d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{12} + 1296t^{10} + 432t^{8} - 38016t^{6} + 112320t^{4} -
89856t^{2} - 6912\]
\[B(t) = 432t^{18} - 7776t^{16} + 114048t^{14} - 1016064t^{12} + 4561920t^{10} -
9621504t^{8} + 6580224t^{6} + 5640192t^{4} - 7630848t^{2} + 221184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 1667x + 72764$, with conductor $147$ |
| Generic density of odd order reductions |
$25/224$ |