## The modular curve $X_{47}$

Curve name $X_{47}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 2 & 3 \end{matrix}\right]$
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 $X_{47}$ minimally covers $X_{11}$
Curves that minimally cover $X_{47}$ $X_{63}$, $X_{64}$, $X_{68}$, $X_{70}$, $X_{74}$, $X_{97}$, $X_{132}$, $X_{133}$, $X_{135}$, $X_{138}$
Curves that minimally cover $X_{47}$ and have infinitely many rational points. $X_{63}$, $X_{64}$, $X_{68}$, $X_{70}$, $X_{74}$, $X_{97}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{47}) = \mathbb{Q}(f_{47}), f_{11} = \frac{2}{f_{47}^{2}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 375x + 6250$, with conductor $1800$
Generic density of odd order reductions $335/1344$