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

Curve name $X_{36p}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{36}$
Meaning/Special name
Chosen covering $X_{36}$
Curves that $X_{36p}$ minimally covers
Curves that minimally cover $X_{36p}$
Curves that minimally cover $X_{36p}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{6} - 864t^{5} + 13824t^{3} + 25920t^{2} - 13824t - 27648$ $B(t) = 432t^{9} + 5184t^{8} + 10368t^{7} - 96768t^{6} - 445824t^{5} - 41472t^{4} + 2515968t^{3} + 3649536t^{2} + 1327104t + 1769472$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 46845x + 23238625$, with conductor $990$
Generic density of odd order reductions $83/672$