## The modular curve $X_{37d}$

Curve name $X_{37d}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{9}$
Meaning/Special name
Chosen covering $X_{37}$
Curves that $X_{37d}$ minimally covers
Curves that minimally cover $X_{37d}$
Curves that minimally cover $X_{37d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -3096576t^{8} - 8847360t^{7} - 10395648t^{6} - 6856704t^{5} - 2944512t^{4} - 857088t^{3} - 162432t^{2} - 17280t - 756$ $B(t) = 1925185536t^{12} + 7813988352t^{11} + 13419675648t^{10} + 12669419520t^{9} + 7001800704t^{8} + 2027814912t^{7} - 253476864t^{5} - 109403136t^{4} - 24744960t^{3} - 3276288t^{2} - 238464t - 7344$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 265x - 1575$, with conductor $640$
Generic density of odd order reductions $401/1792$