## The modular curve $X_{8a}$

Curve name $X_{8a}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 2 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $6$ $X_{8}$
Meaning/Special name
Chosen covering $X_{8}$
Curves that $X_{8a}$ minimally covers
Curves that minimally cover $X_{8a}$
Curves that minimally cover $X_{8a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{6} - 27t^{4} + 135t^{2} - 81$ $B(t) = 54t^{9} - 648t^{7} + 1620t^{5} - 1512t^{3} + 486t$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 4167x - 92759$, with conductor $450$
Generic density of odd order reductions $513/3584$