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

Curve name $X_{38a}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 2 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{8d}$
Meaning/Special name
Chosen covering $X_{38}$
Curves that $X_{38a}$ minimally covers
Curves that minimally cover $X_{38a}$
Curves that minimally cover $X_{38a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{8} + 162t^{6} - 432t^{4} + 648t^{2} - 432$ $B(t) = 54t^{12} - 486t^{10} + 1296t^{8} - 5184t^{4} + 7776t^{2} - 3456$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 931x - 10290$, with conductor $392$
Generic density of odd order reductions $643/5376$