## The modular curve $X_{20b}$

Curve name $X_{20b}$
Index $16$
Level $4$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 3 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 1 & 0 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $1$ $X_{1}$
Meaning/Special name
Chosen covering $X_{20}$
Curves that $X_{20b}$ minimally covers
Curves that minimally cover $X_{20b}$
Curves that minimally cover $X_{20b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{8} + 648t^{7} - 4212t^{6} - 2376t^{5} + 60102t^{4} + 79704t^{3} - 105732t^{2} - 235224t - 107811$ $B(t) = 54t^{12} - 1944t^{11} + 24300t^{10} - 97848t^{9} - 251262t^{8} + 1722384t^{7} + 4821768t^{6} - 8697456t^{5} - 64323558t^{4} - 140447736t^{3} - 157012020t^{2} - 90561240t - 21346578$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 + 4x - 1$, with conductor $162$
Generic density of odd order reductions $71/168$