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// d2.cc: program for 2nd descent from a phi-descent homogeneous space
//////////////////////////////////////////////////////////////////////////
//
// 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/marith.h>
#include <eclib/unimod.h>
#include <eclib/points.h>
#include <eclib/mquartic.h>
#include <eclib/transform.h>
#include <eclib/msoluble.h>
#include <eclib/qc.h>
#include <eclib/quadratic.h>
#include <eclib/conic.h>
#include <eclib/minim.h>
#include <eclib/reduce.h>
#include <eclib/sqfdiv.h>
#include <eclib/desc2.h>
bigcomplex roots[4];
int getquartic(quartic& g); // special version for a*x^4+c*x^2+e quartics
int main()
{
#ifdef NTL_ALL
set_precision("Enter number of decimal places");
#endif
cin.flags( cin.flags() | ios::dec ); //force decimal input (bug fix)
bigint zero; zero=0;
int alldesc=0, verb=0, selmer_only=0;
cout << "Verbose? "; cin >> verb;
initprimes("PRIMES",0);
double hlim=8;
cout << "Limit on height? "; cin >> hlim;
cout << "Stop after first point found? "; cin >> alldesc; alldesc=1-alldesc;
cout << "Selmer only (0/1: if 1, just tests whether second descent possible)? ";
cin >> selmer_only;
quartic g;
while (getquartic(g))
{
bigint I = g.getI(), J=g.getJ();
Curvedata IJ_curve(zero,zero,zero,-27*I,-27*J,0);
bigint tr_u,tr_r,tr_s,tr_t;
Curvedata E = IJ_curve.minimalize(tr_u,tr_r,tr_s,tr_t);
bigint d1=g.geta(), c=g.getcc(), d2=g.gete();
bigint d = d1*d2, cdash = -2*c, ddash = sqr(c)-4*d;
vector<bigint> plist = pdivs(6*d*ddash);
vector<bigint> supp = support(ddash);
long mask=0;
Curvedata E_cd(zero,c,zero,d,zero);
cout << "cd-curve (nearer):\t" << (Curve)E_cd << endl;
Curvedata E_cd_dash(zero,cdash,zero,ddash,zero);
cout << "cd-dash-curve (further):\t" << (Curve)E_cd_dash << endl;
Point P(&E_cd);
Point Q(&E_cd_dash);
cout << "I = " << I << ", J = " << J << "\n";
cout << "Minimal model for Jacobian: " << (Curve)E << endl;
cout << "Checking local solublity at primes " << plist << ":\n";
int i, els, els1; bigint two; two=2;
els = els1 = qpsoluble(g,two);
if(!els1) cout << "Not locally soluble at p = 2\n";
for (i=1; i<plist.size(); i++)
{
els1=new_qpsoluble(g,plist[i]);
if(!els1) cout << "Not locally soluble at p = "<<plist[i]<<"\n";
els = els&els1;
}
if(!els) continue;
cout << "Everywhere locally soluble.\n";
cout<<"------------------------------------------\n";
bigint x,y,z,x2,xz,z2;
int res = desc2(c,d1,d2,plist,supp,supp,mask,hlim,x,y,z,verb,selmer_only,alldesc);
cout << "\nRESULTS\n\n";
switch(res)
{
case 1:
cout<<"Quartic has rational point "; show_xyz(x,y,z);cout<<endl;
xz=x*z; x2=x*x; z2=z*z;
P = Point(&E_cd, d1*x2*z, d1*x*y, z*z2);
cout << "Point on (c,d) curve = " << P << "\n";
cout << "height = " << height(P)<< "\n";
Q = Point(&E_cd_dash,y*y*xz,y*(d1*x2*x2-d2*z2*z2),pow(xz,3));
cout << "Point on (c',d') curve = " << Q << "\n";
cout << "height = " << height(Q)<< "\n\n";
break;
case -1:
cout<<"Quartic has no rational point (no ELS descendents)\n\n";
break;
case 0:
default:
cout<<"Undecided: quartic has ELS descendents but no rational\n";
cout<<"points were found on any.\n\n";
}
}
}
int getquartic(quartic& g) // special version for a*x^4+c*x^2+e quartics
{
bigint a, b, c, d, e;
cout << "Enter quartic coefficients (a,0,c,0,e) or just a c e " << endl;
char ch; cin>>ch;
if(ch=='(') cin>>a>>ch>>b>>ch>>c>>ch>>d>>ch>>e>>ch;
else
{
cin.putback(ch);
cin >> a >> c >> e;
b=0; d=0;
}
if (sign(a)==0) return 0;
g=quartic(a,b,c,d,e);
return 1;
}
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