## The modular curve $X_{38}$

Curve name $X_{38}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 6 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $6$ $X_{8}$
Meaning/Special name
Chosen covering $X_{8}$
Curves that $X_{38}$ minimally covers $X_{8}$, $X_{14}$, $X_{16}$
Curves that minimally cover $X_{38}$ $X_{62}$, $X_{66}$, $X_{38a}$, $X_{38b}$, $X_{38c}$, $X_{38d}$
Curves that minimally cover $X_{38}$ and have infinitely many rational points. $X_{62}$, $X_{66}$, $X_{38a}$, $X_{38b}$, $X_{38c}$, $X_{38d}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{38}) = \mathbb{Q}(f_{38}), f_{8} = f_{38}^{2} - 1$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 442x - 2784$, with conductor $350$
Generic density of odd order reductions $513/3584$