## The modular curve $X_{38b}$

Curve name $X_{38b}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 6 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{8b}$
Meaning/Special name
Chosen covering $X_{38}$
Curves that $X_{38b}$ minimally covers
Curves that minimally cover $X_{38b}$
Curves that minimally cover $X_{38b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{8} + 648t^{6} - 1728t^{4} + 2592t^{2} - 1728$ $B(t) = 432t^{12} - 3888t^{10} + 10368t^{8} - 41472t^{4} + 62208t^{2} - 27648$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 3724x - 82320$, with conductor $3136$
Generic density of odd order reductions $289/1792$