## The modular curve $X_{9a}$

Curve name $X_{9a}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 6 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 2 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{9}$
Meaning/Special name
Chosen covering $X_{9}$
Curves that $X_{9a}$ minimally covers
Curves that minimally cover $X_{9a}$
Curves that minimally cover $X_{9a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{6} - 4752t^{4} - 276480t^{2} - 5308416$ $B(t) = 54t^{9} + 14256t^{7} + 1410048t^{5} + 61931520t^{3} + 1019215872t$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 33x + 62$, with conductor $100$
Generic density of odd order reductions $2659/10752$