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

Curve name $X_{8c}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
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 $X_{8c}$ minimally covers
Curves that minimally cover $X_{8c}$
Curves that minimally cover $X_{8c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{2} - 81$ $B(t) = 54t^{3} - 486t$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 19x + 26$, with conductor $30$
Generic density of odd order reductions $513/3584$