## The modular curve $X_{109}$

Curve name $X_{109}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 14 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$
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 $X_{109}$ minimally covers $X_{45}$
Curves that minimally cover $X_{109}$ $X_{209}$, $X_{220}$, $X_{311}$, $X_{320}$, $X_{382}$, $X_{401}$
Curves that minimally cover $X_{109}$ and have infinitely many rational points. $X_{209}$, $X_{220}$, $X_{320}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{109}) = \mathbb{Q}(f_{109}), f_{45} = -2f_{109}^{2} - 4$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 2000x - 34375$, with conductor $700$
Generic density of odd order reductions $85091/344064$