| Curve name |
$X_{67}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 4 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 6 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{24}$ |
| Curves that $X_{67}$ minimally covers |
$X_{24}$, $X_{28}$, $X_{43}$ |
| Curves that minimally cover $X_{67}$ |
$X_{255}$, $X_{256}$, $X_{67a}$, $X_{67b}$, $X_{67c}$, $X_{67d}$ |
| Curves that minimally cover $X_{67}$ and have infinitely many rational
points. |
$X_{67a}$, $X_{67b}$, $X_{67c}$, $X_{67d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{67}) = \mathbb{Q}(f_{67}), f_{24} =
\frac{-2}{f_{67}^{2}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 63x + 162$, with conductor $360$ |
| Generic density of odd order reductions |
$13/84$ |