## The modular curve $X_{42}$

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
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{10}$
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 $\mathbb{P}^{1}, \mathbb{Q}(X_{42}) = \mathbb{Q}(f_{42}), f_{10} = \frac{f_{42}^{2} - 8}{f_{42}^{2} + 8f_{42} + 8}$
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 $2659/10752$