## The modular curve $X_{112}$

Curve name $X_{112}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 10 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 13 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 12 & 15 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $12$ $X_{45}$
Meaning/Special name
Chosen covering $X_{45}$
Curves that $X_{112}$ minimally covers $X_{45}$
Curves that minimally cover $X_{112}$ $X_{210}$, $X_{216}$, $X_{220}$, $X_{231}$, $X_{310}$, $X_{323}$, $X_{326}$, $X_{351}$, $X_{361}$, $X_{385}$, $X_{396}$, $X_{403}$
Curves that minimally cover $X_{112}$ and have infinitely many rational points. $X_{210}$, $X_{216}$, $X_{220}$, $X_{231}$, $X_{323}$, $X_{326}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{112}) = \mathbb{Q}(f_{112}), f_{45} = -f_{112}^{2} + 4$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 275x + 10933$, with conductor $4606$
Generic density of odd order reductions $85091/344064$