| Curve name |
$X_{42}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{10}$ |
| Curves that $X_{42}$ minimally covers |
$X_{10}$, $X_{16}$, $X_{18}$ |
| Curves that minimally cover $X_{42}$ |
$X_{108}$, $X_{42a}$, $X_{42b}$, $X_{42c}$, $X_{42d}$ |
| Curves that minimally cover $X_{42}$ and have infinitely many rational
points. |
$X_{108}$, $X_{42a}$, $X_{42b}$, $X_{42c}$, $X_{42d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{42}) = \mathbb{Q}(f_{42}), f_{10} =
\frac{f_{42}^{2} - 8}{f_{42}^{2} + 8f_{42} + 8}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 49x + 49$, with conductor $4480$ |
| Generic density of odd order reductions |
$2659/10752$ |