Elliptic Curves with trace t = 1 [Practice]

So here comes the practical aspect of the previous post about elliptic curves with trace $1$.

I use SAGE for practical demonstration. You can use the online notebook functionality of it, e.g., under sagenb.org or nt.sagenb.org.

I use the curve $y^2 = x^3 + x + 4$, which has $19$ points over $\mathbb{F}_{19}$ and hence has trace equal to one.

--- Input
p = 19;
K = GF(19);
E = EllipticCurve(K,[1,4]); #y^2 = x^3 + x + 4
print "E is: ",E;
print "#E[K] = ",E.count_points();  
--- Output
E is:  Elliptic Curve defined by y^2 = x^3 + x + 4 over Finite Field of size 19
#E[GF(19)] =  19