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

Curve name $X_{13g}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{13}$
Curves that $X_{13g}$ minimally covers
Curves that minimally cover $X_{13g}$
Curves that minimally cover $X_{13g}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{4} + 1728t^{3} - 1728t^{2} - 82944t + 331776$ $B(t) = -432t^{6} + 10368t^{5} - 51840t^{4} - 525312t^{3} + 5971968t^{2} - 15925248t$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 69x - 194$, with conductor $30$
Generic density of odd order reductions $513/3584$