## The modular curve $X_{107}$

Curve name $X_{107}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 5 & 15 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 10 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $12$ $X_{39}$
Meaning/Special name
Chosen covering $X_{39}$
Curves that $X_{107}$ minimally covers $X_{39}$
Curves that minimally cover $X_{107}$ $X_{214}$, $X_{222}$, $X_{224}$, $X_{237}$, $X_{281}$, $X_{287}$, $X_{307}$, $X_{308}$, $X_{362}$, $X_{372}$, $X_{374}$, $X_{399}$
Curves that minimally cover $X_{107}$ and have infinitely many rational points. $X_{214}$, $X_{222}$, $X_{224}$, $X_{237}$, $X_{281}$, $X_{308}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{107}) = \mathbb{Q}(f_{107}), f_{39} = 8f_{107}^{2} - 4$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 22402x + 539748$, with conductor $3038$
Generic density of odd order reductions $85091/344064$