## The modular curve $X_{36s}$

Curve name $X_{36s}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{36}$
Curves that $X_{36s}$ minimally covers
Curves that minimally cover $X_{36s}$
Curves that minimally cover $X_{36s}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{8} + 216t^{7} - 3456t^{5} + 6480t^{4} + 3456t^{3} - 6912t^{2}$ $B(t) = -54t^{12} + 648t^{11} - 1296t^{10} - 12096t^{9} + 55728t^{8} - 5184t^{7} - 314496t^{6} + 456192t^{5} - 165888t^{4} + 221184t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 106425x - 13307459$, with conductor $990$
Generic density of odd order reductions $83/672$