## The modular curve $X_{19}$

Curve name $X_{19}$
Index $6$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $3$ $X_{6}$
Meaning/Special name Elliptic curves that acquire full $2$-torsion over $\mathbb{Q}(\sqrt{-2})$
Chosen covering $X_{6}$
Curves that $X_{19}$ minimally covers $X_{4}$, $X_{6}$
Curves that minimally cover $X_{19}$ $X_{29}$, $X_{48}$
Curves that minimally cover $X_{19}$ and have infinitely many rational points. $X_{29}$, $X_{48}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{19}) = \mathbb{Q}(f_{19}), f_{6} = \frac{48f_{19}^{2} - 32}{f_{19}^{2} + 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + 4x + 20$, with conductor $66$
Generic density of odd order reductions $5123/21504$