## The modular curve $X_{110}$

Curve name $X_{110}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 14 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 6 & 11 \end{matrix}\right]$
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 $X_{110}$ minimally covers $X_{45}$
Curves that minimally cover $X_{110}$ $X_{209}$, $X_{216}$, $X_{218}$, $X_{231}$, $X_{312}$, $X_{321}$, $X_{325}$, $X_{349}$, $X_{367}$, $X_{383}$, $X_{387}$, $X_{397}$
Curves that minimally cover $X_{110}$ and have infinitely many rational points. $X_{209}$, $X_{216}$, $X_{218}$, $X_{231}$, $X_{312}$, $X_{349}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{110}) = \mathbb{Q}(f_{110}), f_{45} = 8f_{110}^{2} - 4$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 5038x + 62292$, with conductor $3038$
Generic density of odd order reductions $85091/344064$