read NormalBasis;

f :=
x^3*y^5-x^3*(-5+16*x)*y^4+2*x^3*(-37*x+43*x^2+13)*y^3+2*x^2*(50*x+232*x^3-8-\
135*x^2-180*x^4+40*x^5)*y^2+(-834*x^4-160*x^8-30*x^2+1516*x^5+255*x^3-80*x^9+
868*x^7-8*x-1648*x^6+2)*y+3-13*x-4*x^2+215*x^3+1608*x^7+80*x^9+64*x^10-592*x^8
+32*x^11-2100*x^6+1709*x^5-845*x^4;

n := degree(f,y);

alpha := RootOf(f,y):

V := Candidates(f,x,y);

beta := select(has,V,y)[1];
beta :=  subs(y=alpha, beta):
m_beta := evala(Norm(y - beta));