## The modular curve $X_{40}$

Curve name $X_{40}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{12}$
Meaning/Special name
Chosen covering $X_{12}$
Curves that $X_{40}$ minimally covers $X_{12}$, $X_{16}$, $X_{17}$
Curves that minimally cover $X_{40}$ $X_{126}$, $X_{145}$, $X_{40a}$, $X_{40b}$, $X_{40c}$, $X_{40d}$
Curves that minimally cover $X_{40}$ and have infinitely many rational points. $X_{40a}$, $X_{40b}$, $X_{40c}$, $X_{40d}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{40}) = \mathbb{Q}(f_{40}), f_{12} = \frac{8f_{40}^{2} - 1}{f_{40}^{2} + f_{40} + \frac{1}{8}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 318x + 1640$, with conductor $16128$
Generic density of odd order reductions $3331/10752$