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

Curve name $X_{40d}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 3 \\ 14 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 9 \end{matrix}\right]$
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 $X_{40d}$ minimally covers
Curves that minimally cover $X_{40d}$
Curves that minimally cover $X_{40d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -70778880t^{12} - 169869312t^{11} - 130940928t^{10} - 21233664t^{9} + 15261696t^{8} + 5308416t^{7} + 552960t^{6} + 663552t^{5} + 238464t^{4} - 41472t^{3} - 31968t^{2} - 5184t - 270$ $B(t) = 202937204736t^{18} + 652298158080t^{17} + 641426522112t^{16} - 97844723712t^{15} - 625119068160t^{14} - 411763212288t^{13} - 15174991872t^{12} + 102431195136t^{11} + 45312638976t^{10} - 5664079872t^{8} - 1600487424t^{7} + 29638656t^{6} + 100528128t^{5} + 19077120t^{4} + 373248t^{3} - 305856t^{2} - 38880t - 1512$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 1731x - 20257$, with conductor $12544$
Generic density of odd order reductions $106595/344064$