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

Curve name $X_{42a}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 14 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 10 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 14 & 7 \end{matrix}\right]$
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 $X_{42a}$ minimally covers
Curves that minimally cover $X_{42a}$
Curves that minimally cover $X_{42a}$ 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 $y^2 = x^3 + x^2 - 1533x + 29563$, with conductor $3200$
Generic density of odd order reductions $85091/344064$