| Curve name |
$X_{85n}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{85}$ |
| Curves that $X_{85n}$ minimally covers |
|
| Curves that minimally cover $X_{85n}$ |
|
| Curves that minimally cover $X_{85n}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{12} + 1728t^{10} - 4320t^{8} - 44928t^{6} + 260928t^{4} -
373248t^{2} - 27648\]
\[B(t) = -432t^{18} + 10368t^{16} - 160704t^{14} + 1693440t^{12} -
10554624t^{10} + 35500032t^{8} - 52835328t^{6} - 331776t^{4} + 59719680t^{2} -
1769472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 232470x + 37012950$, with conductor $2070$ |
| Generic density of odd order reductions |
$193/1792$ |