## The modular curve $X_{39}$

Curve name $X_{39}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$
Meaning/Special name
Chosen covering $X_{11}$
Curves that $X_{39}$ minimally covers $X_{11}$
Curves that minimally cover $X_{39}$ $X_{63}$, $X_{66}$, $X_{80}$, $X_{83}$, $X_{90}$, $X_{91}$, $X_{95}$, $X_{97}$, $X_{105}$, $X_{106}$, $X_{107}$, $X_{124}$, $X_{128}$, $X_{143}$, $X_{144}$, $X_{147}$, $X_{160}$, $X_{161}$, $X_{165}$, $X_{166}$
Curves that minimally cover $X_{39}$ and have infinitely many rational points. $X_{63}$, $X_{66}$, $X_{80}$, $X_{83}$, $X_{90}$, $X_{91}$, $X_{95}$, $X_{97}$, $X_{105}$, $X_{106}$, $X_{107}$, $X_{124}$, $X_{165}$, $X_{166}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{39}) = \mathbb{Q}(f_{39}), f_{11} = \frac{-64}{f_{39}^{2} - 8}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 1617x + 18666$, with conductor $1575$
Generic density of odd order reductions $2659/10752$