## The modular curve $X_{40b}$

Curve name $X_{40b}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 6 & 11 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 14 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{12}$ $8$ $12$ $X_{40}$
Meaning/Special name
Chosen covering $X_{40}$
Curves that $X_{40b}$ minimally covers
Curves that minimally cover $X_{40b}$
Curves that minimally cover $X_{40b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -17280t^{4} - 6912t^{3} - 864t^{2} - 864t - 270$ $B(t) = 774144t^{6} + 165888t^{5} - 373248t^{4} - 235008t^{3} - 46656t^{2} + 2592t + 1512$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 35x - 49$, with conductor $1792$
Generic density of odd order reductions $106595/344064$