## The modular curve $X_{119}$

Curve name $X_{119}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{36}$
Meaning/Special name
Chosen covering $X_{36}$
Curves that $X_{119}$ minimally covers $X_{36}$
Curves that minimally cover $X_{119}$ $X_{213}$, $X_{215}$, $X_{219}$, $X_{233}$, $X_{338}$, $X_{344}$, $X_{119a}$, $X_{119b}$, $X_{119c}$, $X_{119d}$, $X_{119e}$, $X_{119f}$, $X_{119g}$, $X_{119h}$, $X_{119i}$, $X_{119j}$, $X_{119k}$, $X_{119l}$, $X_{119m}$, $X_{119n}$, $X_{119o}$, $X_{119p}$
Curves that minimally cover $X_{119}$ and have infinitely many rational points. $X_{213}$, $X_{215}$, $X_{219}$, $X_{233}$, $X_{119a}$, $X_{119b}$, $X_{119c}$, $X_{119d}$, $X_{119e}$, $X_{119f}$, $X_{119g}$, $X_{119h}$, $X_{119i}$, $X_{119j}$, $X_{119k}$, $X_{119l}$, $X_{119m}$, $X_{119n}$, $X_{119o}$, $X_{119p}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{119}) = \mathbb{Q}(f_{119}), f_{36} = -f_{119}^{2} - 4$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 4851x - 128840$, with conductor $1989$
Generic density of odd order reductions $83/672$