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

Curve name $X_{42d}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{10d}$
Meaning/Special name
Chosen covering $X_{42}$
Curves that $X_{42d}$ minimally covers
Curves that minimally cover $X_{42d}$
Curves that minimally cover $X_{42d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = 54t^{8} - 432t^{7} - 4320t^{6} + 3456t^{5} + 62208t^{4} + 27648t^{3} - 276480t^{2} - 221184t + 221184$ $B(t) = 540t^{12} + 5184t^{11} - 5184t^{10} - 138240t^{9} - 20736t^{8} + 1658880t^{7} - 13271040t^{5} + 1327104t^{4} + 70778880t^{3} + 21233664t^{2} - 169869312t - 141557760$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 15x - 25$, with conductor $640$
Generic density of odd order reductions $401/1792$