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

Curve name $X_{42b}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 6 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{10b}$
Meaning/Special name
Chosen covering $X_{42}$
Curves that $X_{42b}$ minimally covers
Curves that minimally cover $X_{42b}$
Curves that minimally cover $X_{42b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = 216t^{8} - 1728t^{7} - 17280t^{6} + 13824t^{5} + 248832t^{4} + 110592t^{3} - 1105920t^{2} - 884736t + 884736$ $B(t) = 4320t^{12} + 41472t^{11} - 41472t^{10} - 1105920t^{9} - 165888t^{8} + 13271040t^{7} - 106168320t^{5} + 10616832t^{4} + 566231040t^{3} + 169869312t^{2} - 1358954496t - 1132462080$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 61x + 261$, with conductor $640$
Generic density of odd order reductions $1427/5376$