"Source: Text/Geometry/CrvCon.text";
"Line: 1538";
"Date: Thu Sep 25 14:47:45 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Geometry/CrvCon.text, line: 1538
// Example: H130E13 ()
print "Example: H130E13";
ei := GetEchoInput();
SetEchoInput(true);
P2<x,y,z> := ProjectiveSpace(Rationals(), 2);
C0 := Conic(P2, 2*x^2 + x*y + 5*y^2 - 37*z^2);
HasRationalPoint(C0);
assert not $1;
BadPrimes(C0);
A := QuaternionAlgebra(C0);
RamifiedPrimes(A);
B := Basis(MaximalOrder(A));
B;
m1 := Automorphism(C0, A!B[2]);
m1 : Minimal;
m2 := Automorphism(C0, A!B[3]);
m2 : Minimal;
m3 := Automorphism(C0, A!B[4]);
m3 : Minimal;
[ Trace(B[2]), Trace(B[3]), Trace(B[4]) ];
m1 * m1 : Minimal;
m2 * m2 : Minimal;
m3 * m3 : Minimal;
C1 := Conic(P2, 2*x^2 + x*y + 5*y^2 - z^2);
HasRationalPoint(C1);
assert not $1;
C2 := Conic(P2, 2*x^2 + x*y + 5*y^2 - 2*z^2);
HasRationalPoint(C2);
assert $1;
RationalPoint(C2);
P<t> := PolynomialRing(RationalField());
L<a> := NumberField(t^2 - 74);
p := C0(L)![1, 0, a/37];
m1(p);
m2(p);
m3(p);
SetEchoInput(ei);
