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

Curve name $X_{67d}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 6 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{24}$
Meaning/Special name
Chosen covering $X_{67}$
Curves that $X_{67d}$ minimally covers
Curves that minimally cover $X_{67d}$
Curves that minimally cover $X_{67d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{16} - 1296t^{12} - 6912t^{8} - 20736t^{4} - 27648$ $B(t) = 432t^{24} + 7776t^{20} + 41472t^{16} - 663552t^{8} - 1990656t^{4} - 1769472$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 700x - 6000$, with conductor $1600$
Generic density of odd order reductions $419/2688$