## The modular curve $X_{42c}$

Curve name $X_{42c}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{10}$ $8$ $12$ $X_{42}$
Meaning/Special name
Chosen covering $X_{42}$
Curves that $X_{42c}$ minimally covers
Curves that minimally cover $X_{42c}$
Curves that minimally cover $X_{42c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = 54t^{12} - 6048t^{10} - 41472t^{9} - 72576t^{8} + 331776t^{7} + 1880064t^{6} + 2654208t^{5} - 4644864t^{4} - 21233664t^{3} - 24772608t^{2} + 14155776$ $B(t) = -540t^{18} - 11664t^{17} - 95904t^{16} - 311040t^{15} + 414720t^{14} + 6469632t^{13} + 15704064t^{12} - 19906560t^{11} - 145981440t^{10} + 1167851520t^{8} + 1274019840t^{7} - 8040480768t^{6} - 26499612672t^{5} - 13589544960t^{4} + 81537269760t^{3} + 201125265408t^{2} + 195689447424t + 72477573120$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 383x + 3887$, with conductor $3200$
Generic density of odd order reductions $85091/344064$