| Curve name |
$X_{85b}$ |
| 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 \\ 8 & 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 $X_{85b}$ minimally covers |
|
| Curves that minimally cover $X_{85b}$ |
|
| Curves that minimally cover $X_{85b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{12} + 324t^{10} + 108t^{8} - 9504t^{6} + 28080t^{4} - 22464t^{2}
- 1728\]
\[B(t) = 54t^{18} - 972t^{16} + 14256t^{14} - 127008t^{12} + 570240t^{10} -
1202688t^{8} + 822528t^{6} + 705024t^{4} - 953856t^{2} + 27648\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - 18131x - 889659$, with conductor $1470$ |
| Generic density of odd order reductions |
$193/1792$ |