| Curve name |
$X_{67d}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 6 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{67}$ |
| Curves that $X_{67d}$ minimally covers |
|
| Curves that minimally cover $X_{67d}$ |
|
| Curves that minimally cover $X_{67d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} - 1296t^{12} - 6912t^{8} - 20736t^{4} - 27648\]
\[B(t) = 432t^{24} + 7776t^{20} + 41472t^{16} - 663552t^{8} - 1990656t^{4} -
1769472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 700x - 6000$, with conductor $1600$ |
| Generic density of odd order reductions |
$419/2688$ |