## The modular curve $X_{49}$

Curve name $X_{49}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$
Meaning/Special name
Chosen covering $X_{11}$
Curves that $X_{49}$ minimally covers $X_{11}$
Curves that minimally cover $X_{49}$ $X_{70}$, $X_{71}$, $X_{76}$, $X_{95}$, $X_{130}$, $X_{137}$
Curves that minimally cover $X_{49}$ and have infinitely many rational points. $X_{70}$, $X_{71}$, $X_{76}$, $X_{95}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{49}) = \mathbb{Q}(f_{49}), f_{11} = \frac{64}{f_{49}^{2} + 8}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 + 292x + 9588$, with conductor $300$
Generic density of odd order reductions $2659/10752$