## The modular curve $X_{13}$

Curve name $X_{13}$
Index $6$
Level $4$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$
Meaning/Special name Elliptic curves with a cyclic $4$-isogeny defined over $\mathbb{Q}$
Chosen covering $X_{6}$
Curves that $X_{13}$ minimally covers $X_{6}$
Curves that minimally cover $X_{13}$ $X_{25}$, $X_{27}$, $X_{32}$, $X_{33}$, $X_{34}$, $X_{36}$, $X_{44}$, $X_{48}$, $X_{13a}$, $X_{13b}$, $X_{13c}$, $X_{13d}$, $X_{13e}$, $X_{13f}$, $X_{13g}$, $X_{13h}$
Curves that minimally cover $X_{13}$ and have infinitely many rational points. $X_{25}$, $X_{27}$, $X_{32}$, $X_{33}$, $X_{34}$, $X_{36}$, $X_{44}$, $X_{48}$, $X_{13a}$, $X_{13b}$, $X_{13c}$, $X_{13d}$, $X_{13e}$, $X_{13f}$, $X_{13g}$, $X_{13h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{13}) = \mathbb{Q}(f_{13}), f_{6} = -f_{13}^{2} + 48$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 23x - 34$, with conductor $315$
Generic density of odd order reductions $1/7$