## The modular curve $X_{67b}$

Curve name $X_{67b}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 2 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{24d}$
Meaning/Special name
Chosen covering $X_{67}$
Curves that $X_{67b}$ minimally covers
Curves that minimally cover $X_{67b}$
Curves that minimally cover $X_{67b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{16} - 324t^{12} - 1728t^{8} - 5184t^{4} - 6912$ $B(t) = 54t^{24} + 972t^{20} + 5184t^{16} - 82944t^{8} - 248832t^{4} - 221184$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 175x - 750$, with conductor $200$
Generic density of odd order reductions $103/672$