| Curve name |
$X_{84o}$ |
| 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} 1 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{84}$ |
| Curves that $X_{84o}$ minimally covers |
|
| Curves that minimally cover $X_{84o}$ |
|
| Curves that minimally cover $X_{84o}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{12} - 216t^{10} - 270t^{8} + 1404t^{6} + 4077t^{4} + 2916t^{2} -
108\]
\[B(t) = 54t^{18} + 648t^{16} + 5022t^{14} + 26460t^{12} + 82458t^{10} +
138672t^{8} + 103194t^{6} - 324t^{4} - 29160t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 2892x + 512624$, with conductor $2880$ |
| Generic density of odd order reductions |
$25/224$ |