## The modular curve $X_{36}$

Curve name $X_{36}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name $X_{0}(8)$
Chosen covering $X_{13}$
Curves that $X_{36}$ minimally covers $X_{13}$
Curves that minimally cover $X_{36}$ $X_{75}$, $X_{78}$, $X_{84}$, $X_{85}$, $X_{86}$, $X_{92}$, $X_{96}$, $X_{102}$, $X_{117}$, $X_{118}$, $X_{119}$, $X_{120}$, $X_{121}$, $X_{122}$, $X_{159}$, $X_{168}$, $X_{36a}$, $X_{36b}$, $X_{36c}$, $X_{36d}$, $X_{36e}$, $X_{36f}$, $X_{36g}$, $X_{36h}$, $X_{36i}$, $X_{36j}$, $X_{36k}$, $X_{36l}$, $X_{36m}$, $X_{36n}$, $X_{36o}$, $X_{36p}$, $X_{36q}$, $X_{36r}$, $X_{36s}$, $X_{36t}$
Curves that minimally cover $X_{36}$ and have infinitely many rational points. $X_{75}$, $X_{78}$, $X_{84}$, $X_{85}$, $X_{86}$, $X_{92}$, $X_{96}$, $X_{102}$, $X_{117}$, $X_{118}$, $X_{119}$, $X_{120}$, $X_{121}$, $X_{122}$, $X_{36a}$, $X_{36b}$, $X_{36c}$, $X_{36d}$, $X_{36e}$, $X_{36f}$, $X_{36g}$, $X_{36h}$, $X_{36i}$, $X_{36j}$, $X_{36k}$, $X_{36l}$, $X_{36m}$, $X_{36n}$, $X_{36o}$, $X_{36p}$, $X_{36q}$, $X_{36r}$, $X_{36s}$, $X_{36t}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{36}) = \mathbb{Q}(f_{36}), f_{13} = -f_{36}^{2} + 8$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 2385x - 44240$, with conductor $3465$
Generic density of odd order reductions $5/42$