| Curve name |
$X_{84k}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{84}$ |
| Curves that $X_{84k}$ minimally covers |
|
| Curves that minimally cover $X_{84k}$ |
|
| Curves that minimally cover $X_{84k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{8} - 108t^{6} + 270t^{4} + 756t^{2} - 27\]
\[B(t) = 54t^{12} + 324t^{10} + 2430t^{8} + 7560t^{6} + 5346t^{4} - 3564t^{2} -
54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 80x + 2400$, with conductor $240$ |
| Generic density of odd order reductions |
$9/112$ |