"Source: Text/Geometry/Lseries.text";
"Line: 646";
"Date: Thu Sep 25 14:47:45 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Geometry/Lseries.text, line: 646
// Example: H139E13 ()
print "Example: H139E13";
ei := GetEchoInput();
SetEchoInput(true);
K<i>:=CyclotomicField(4);
R<x>:=PolynomialRing(K);
C:=HyperellipticCurve((-i-1)*x^3+(-i-1)*x^2-i*x,x^3+x^2+x+1);
L:=LSeries(C: Precision:=8);
LCfRequired(L);                // need this many coefficients
CheckFunctionalEquation(L);    // zero to correct precision
Evaluate(L,1);                 // L(C,1) is zero
Evaluate(L,1: Derivative:=1);  // L'(C,1) is non-zero
f:=HyperellipticPolynomials(SimplifiedModel(C));
J:=Jacobian(HyperellipticCurve(f));
#TwoTorsionSubgroup(J),#TwoSelmerGroup(J);
Points(C: Bound:=10);
SetEchoInput(ei);
