## The modular curve $X_{16}$

Curve name $X_{16}$
Index $6$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $3$ $X_{6}$
Meaning/Special name Elliptic curves with a cyclic $4$-isogeny defined over $\mathbb{Q}(\sqrt{2})$
Chosen covering $X_{6}$
Curves that $X_{16}$ minimally covers $X_{6}$
Curves that minimally cover $X_{16}$ $X_{29}$, $X_{38}$, $X_{40}$, $X_{42}$
Curves that minimally cover $X_{16}$ and have infinitely many rational points. $X_{29}$, $X_{38}$, $X_{40}$, $X_{42}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{16}) = \mathbb{Q}(f_{16}), f_{6} = -2f_{16}^{2} + 48$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + 4x - 6$, with conductor $14$
Generic density of odd order reductions $5123/21504$