| Curve name |
$X_{66e}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 6 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{66}$ |
| Curves that $X_{66e}$ minimally covers |
|
| Curves that minimally cover $X_{66e}$ |
|
| Curves that minimally cover $X_{66e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -81t^{8} - 1296t^{7} - 12096t^{6} - 72576t^{5} - 266112t^{4} -
580608t^{3} - 774144t^{2} - 663552t - 331776\]
\[B(t) = 3888t^{11} + 85536t^{10} + 874368t^{9} + 5474304t^{8} + 23721984t^{7} +
76640256t^{6} + 189775872t^{5} + 350355456t^{4} + 447676416t^{3} +
350355456t^{2} + 127401984t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 817x + 8976$, with conductor $1120$ |
| Generic density of odd order reductions |
$307/2688$ |