## The modular curve $X_{111}$

Curve name $X_{111}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 10 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 13 \end{matrix}\right]$
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 $X_{111}$ minimally covers $X_{45}$
Curves that minimally cover $X_{111}$ $X_{210}$, $X_{218}$, $X_{309}$, $X_{322}$, $X_{375}$, $X_{384}$
Curves that minimally cover $X_{111}$ and have infinitely many rational points. $X_{210}$, $X_{218}$, $X_{309}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{111}) = \mathbb{Q}(f_{111}), f_{45} = f_{111}^{2} + 4$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 7020x + 226368$, with conductor $2016$
Generic density of odd order reductions $85091/344064$