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

Curve name $X_{13a}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 1 \end{matrix}\right]$
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 $X_{13a}$ minimally covers
Curves that minimally cover $X_{13a}$
Curves that minimally cover $X_{13a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{6} + 19008t^{4} - 1105920t^{2} + 21233664$ $B(t) = 432t^{9} - 114048t^{7} + 11280384t^{5} - 495452160t^{3} + 8153726976t$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 333x - 7259$, with conductor $450$
Generic density of odd order reductions $513/3584$