## The modular curve $X_{50}$

Curve name $X_{50}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$
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 $X_{50}$ minimally covers $X_{11}$
Curves that minimally cover $X_{50}$ $X_{65}$, $X_{69}$, $X_{71}$, $X_{73}$, $X_{90}$, $X_{91}$, $X_{113}$, $X_{114}$, $X_{125}$, $X_{131}$, $X_{134}$, $X_{141}$, $X_{152}$, $X_{154}$, $X_{155}$, $X_{156}$, $X_{170}$, $X_{171}$
Curves that minimally cover $X_{50}$ and have infinitely many rational points. $X_{65}$, $X_{69}$, $X_{71}$, $X_{73}$, $X_{90}$, $X_{91}$, $X_{113}$, $X_{114}$, $X_{155}$, $X_{156}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{50}) = \mathbb{Q}(f_{50}), f_{11} = f_{50}^{2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 248453x - 47558454$, with conductor $71148$
Generic density of odd order reductions $2659/10752$