"Source: Text/RepThy/GalRep.text";
"Line: 399";
"Date: Fri Sep 26 12:10:12 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/RepThy/GalRep.text, line: 399
// Example: H58E7 ()
print "Example: H58E7";
ei := GetEchoInput();
SetEchoInput(true);
_<x>:=PolynomialRing(Rationals());
p:=10007;
K:=pAdicField(p,20);
_<X>:=PolynomialRing(K);
C:=HyperellipticCurve(X^5+X+1); C;
k:=ResidueClassField(Integers(K));
_<X>:=PolynomialRing(k);
Ck:=HyperellipticCurve(X^5+X+1); 
ap:=p+1-#Ck; ap;
A:=UnramifiedRepresentation(K,4,1,1-ap*x);
A;
A*A;
EulerFactor(A*A+1/CyclotomicCharacter(K));
SetEchoInput(ei);
