## The modular curve $X_{113}$

Curve name $X_{113}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 3 \\ 12 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 12 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 14 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $12$ $X_{50}$
Meaning/Special name
Chosen covering $X_{50}$
Curves that $X_{113}$ minimally covers $X_{50}$
Curves that minimally cover $X_{113}$ $X_{284}$, $X_{290}$, $X_{317}$, $X_{324}$, $X_{328}$, $X_{348}$
Curves that minimally cover $X_{113}$ and have infinitely many rational points. $X_{284}$, $X_{324}$, $X_{328}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{113}) = \mathbb{Q}(f_{113}), f_{50} = -f_{113}^{2}$
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 785472x - 267747760$, with conductor $2093058$
Generic density of odd order reductions $85091/344064$