## The modular curve $X_{48}$

Curve name $X_{48}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{13}$
Curves that $X_{48}$ minimally covers $X_{13}$, $X_{18}$, $X_{19}$
Curves that minimally cover $X_{48}$ $X_{75}$, $X_{48a}$, $X_{48b}$, $X_{48c}$, $X_{48d}$
Curves that minimally cover $X_{48}$ and have infinitely many rational points. $X_{75}$, $X_{48a}$, $X_{48b}$, $X_{48c}$, $X_{48d}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{48}) = \mathbb{Q}(f_{48}), f_{13} = \frac{8f_{48}^{2} - 16}{f_{48}^{2} + 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 108x + 2074$, with conductor $198$
Generic density of odd order reductions $513/3584$