| Curve name |
$X_{67b}$ |
| 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 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{67}$ |
| Curves that $X_{67b}$ minimally covers |
|
| Curves that minimally cover $X_{67b}$ |
|
| Curves that minimally cover $X_{67b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} - 324t^{12} - 1728t^{8} - 5184t^{4} - 6912\]
\[B(t) = 54t^{24} + 972t^{20} + 5184t^{16} - 82944t^{8} - 248832t^{4} - 221184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 175x - 750$, with conductor $200$ |
| Generic density of odd order reductions |
$103/672$ |