//// given a elliptic curve E : y^2+xy-x^3- a2 x^2-a4 x
///  over F_2^d
///  output : the trace of the frobenius or 2^(Ceiling(d/2)+1)-trace.




agm1:=function(ab2,ab4)
d:=Degree(Parent(ab2));
cc:=2;
prec:=d+5;

P2:=LocalRing(cc,d,prec);
Q2<p,W>:=LocalField(cc,d,prec);
//Q2`SeriesPrinting:=true;
F2<w>,phi:=ResidueClassField(P2);
S<x>:=PolynomialRing(Q2);
Embed(F2,Parent(ab2));

a2:=F2!ab2;
a4:=F2!ab4;

//a2:=w^5+1;
//a4:=w^2+w^5;

E:=EllipticCurve([F2!1,a2,0,a4,0]);
a2:=Q2!a2;
a4:=Q2!a4;

pp:=x^2+(a2+1/4)*x+a4;
rr:=Roots(pp);

if Valuation(rr[1][1]) eq -2 then l0:=-rr[1][1];l1:=rr[2][1]-rr[1][1];
else l0:=-rr[2][1];l1:=rr[1][1]-rr[2][1];end if;

l:=Sqrt(l1/l0);


ll:=l;
for i:=1 to Ceiling(d/2)+5 do
ll:=ChangePrecision(2*Sqrt(ll)/(1+ll),prec);
end for;


a:=1;b:=ll;
for i:=1 to d do
a1:=ChangePrecision((a+b)/2,prec);
b1:=ChangePrecision(Sqrt(a*b),prec);
a:=a1;b:=b1;
end for;

return ChangePrecision(1/a+2^d*a,Ceiling(d/2)+1);
end function;