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

Curve name $X_{36d}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$
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 $X_{36d}$ minimally covers
Curves that minimally cover $X_{36d}$
Curves that minimally cover $X_{36d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{10} + 5184t^{8} - 84672t^{6} + 497664t^{4} - 442368t^{2}$ $B(t) = 432t^{15} - 31104t^{13} + 881280t^{11} - 12192768t^{9} + 80953344t^{7} - 191102976t^{5} - 113246208t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 135558x + 74465716$, with conductor $54450$
Generic density of odd order reductions $643/5376$