| Curve name |
$X_{66d}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 2 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{66}$ |
| Curves that $X_{66d}$ minimally covers |
|
| Curves that minimally cover $X_{66d}$ |
|
| Curves that minimally cover $X_{66d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -324t^{8} - 5184t^{7} - 48384t^{6} - 290304t^{5} - 1064448t^{4} -
2322432t^{3} - 3096576t^{2} - 2654208t - 1327104\]
\[B(t) = 31104t^{11} + 684288t^{10} + 6994944t^{9} + 43794432t^{8} +
189775872t^{7} + 613122048t^{6} + 1518206976t^{5} + 2802843648t^{4} +
3581411328t^{3} + 2802843648t^{2} + 1019215872t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 3268x + 71808$, with conductor $2240$ |
| Generic density of odd order reductions |
$419/2688$ |