| Curve name |
$X_{85e}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 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 \\ 8 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{85}$ |
| Curves that $X_{85e}$ minimally covers |
|
| Curves that minimally cover $X_{85e}$ |
|
| Curves that minimally cover $X_{85e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} + 540t^{14} - 2916t^{12} - 5184t^{10} + 105840t^{8} -
399168t^{6} + 627264t^{4} - 345600t^{2} - 27648\]
\[B(t) = 54t^{24} - 1620t^{22} + 28512t^{20} - 348192t^{18} + 2840832t^{16} -
15054336t^{14} + 50754816t^{12} - 103389696t^{10} + 107039232t^{8} -
7326720t^{6} - 91238400t^{4} + 62373888t^{2} - 1769472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 - 4079480x + 2998519647$, with conductor
$22050$ |
| Generic density of odd order reductions |
$193/1792$ |