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// torsion.cc: program to find & display torsion points
//////////////////////////////////////////////////////////////////////////
//
// Copyright 1990-2012 John Cremona
//
// This file is part of the eclib package.
//
// eclib is free software; you can redistribute it and/or modify it
// under the terms of the GNU General Public License as published by the
// Free Software Foundation; either version 2 of the License, or (at your
// option) any later version.
//
// eclib is distributed in the hope that it will be useful, but WITHOUT
// ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
// FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
// for more details.
//
// You should have received a copy of the GNU General Public License
// along with eclib; if not, write to the Free Software Foundation,
// Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA
//
//////////////////////////////////////////////////////////////////////////
#include <eclib/points.h>
#include <eclib/getcurve.h>
int main(){
cerr<<"Program to find and/or count torsion on a curve.\n\n";
#ifdef MPFP
set_precision("Enter precision in bits");
#endif
cerr<<"Enter 0 for short output (#torsion only)\n";
cerr<<" or 1 for long output (list of the torsion points): ";
int showpoints; cin >> showpoints;
initprimes("PRIMES",0);
vector<bigrational> ai(5);
vector<Point> tor; int ntor, n2;
bigint u, r, s, t, v;
int verb=1;
while (getcurve(ai,verb))
{
Curvedata E0(ai,v);
Curvedata E1 = E0.minimalize(u,r,s,t);
cout<<"Curve ["<<ai[0]<<","<<ai[1]<<","<<ai[2]<<","<<ai[3]<<","<<ai[4]<<"]"<<flush;
tor = torsion_points(E1);
ntor = tor.size();
n2=0;
cout<<" \t has " << ntor << " torsion point(s)\n";
if(showpoints)
{
for(int i=0; i<ntor; i++)
{
Point P = transform(tor[i],&E0,u,r,s,t,1);
cout << scale(P,v,0) << "\t";
if(!P.isvalid()) cout << " --warning: NOT on curve!\n";
else
{
int ord = order(P);
if(ord==2) n2++;
cout << "(order " << ord << ")\n";
}
}
if(n2>2) cout<<"Non-cyclic: C2 x C"<<(ntor/2)<<"\n";
else cout << "Cyclic: C"<<ntor<<"\n";
}
}
} //ends main
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