## The modular curve $X_{13h}$

Curve name $X_{13h}$
Index $12$
Level $4$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$
Meaning/Special name $X_1(4)$
Chosen covering $X_{13}$
Curves that $X_{13h}$ minimally covers
Curves that minimally cover $X_{13h}$
Curves that minimally cover $X_{13h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{4} + 432t^{3} - 432t^{2} - 20736t + 82944$ $B(t) = 54t^{6} - 1296t^{5} + 6480t^{4} + 65664t^{3} - 746496t^{2} + 1990656t$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + x^2 - 146x + 621$, with conductor $33$
Generic density of odd order reductions $5/42$