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

Curve name $X_{36t}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{36}$
Meaning/Special name
Chosen covering $X_{36}$
Curves that $X_{36t}$ minimally covers
Curves that minimally cover $X_{36t}$
Curves that minimally cover $X_{36t}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{6} + 216t^{5} - 3456t^{3} + 6480t^{2} + 3456t - 6912$ $B(t) = 54t^{9} - 648t^{8} + 1296t^{7} + 12096t^{6} - 55728t^{5} + 5184t^{4} + 314496t^{3} - 456192t^{2} + 165888t - 221184$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 2295x - 4595$, with conductor $990$
Generic density of odd order reductions $83/672$