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

Curve name $X_{36q}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
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 $X_{36q}$ minimally covers
Curves that minimally cover $X_{36q}$
Curves that minimally cover $X_{36q}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{8} + 864t^{7} - 13824t^{5} + 25920t^{4} + 13824t^{3} - 27648t^{2}$ $B(t) = 432t^{12} - 5184t^{11} + 10368t^{10} + 96768t^{9} - 445824t^{8} + 41472t^{7} + 2515968t^{6} - 3649536t^{5} + 1327104t^{4} - 1769472t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 1539765x + 735795481$, with conductor $990$
Generic density of odd order reductions $83/672$