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

Curve name $X_{36g}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{36}$
Meaning/Special name
Chosen covering $X_{36}$
Curves that $X_{36g}$ minimally covers
Curves that minimally cover $X_{36g}$
Curves that minimally cover $X_{36g}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{10} + 1296t^{8} - 21168t^{6} + 124416t^{4} - 110592t^{2}$ $B(t) = 54t^{15} - 3888t^{13} + 110160t^{11} - 1524096t^{9} + 10119168t^{7} - 23887872t^{5} - 14155776t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 1815386x - 940703200$, with conductor $53361$
Generic density of odd order reductions $83/672$