/usr/include/eclib/curve.h is in libec-dev 20160101-1.
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//////////////////////////////////////////////////////////////////////////
//
// 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
//
//////////////////////////////////////////////////////////////////////////
// originally adapted from Elliptic.h by Oisin McGuiness
// allow for multiple includes
#ifndef _ELLIPTIC_
#define _ELLIPTIC_
#include "p2points.h"
#include <map>
class Curve; class Curvedata; class CurveRed;
class Point;
class IsogenyClass;
//general test:
int valid_invariants(const bigint& c4, const bigint& c6); //true if valid
void c4c6_to_ai(const bigint& c4, const bigint& c6,
bigint& a1, bigint& a2, bigint& a3, bigint& a4,
bigint& a6,
bigint& b2, bigint& b4, bigint& b6, bigint& b8);
void c4c6_to_ai(const bigint& c4, const bigint& c6,
bigint& a1, bigint& a2, bigint& a3, bigint& a4,
bigint& a6);
void minimise_c4c6(const bigint& c4, const bigint& c6, const bigint& discr,
bigint& newc4, bigint& newc6, bigint& newdiscr, bigint& u);
//
// base class for bare elliptic curve
//
class Curve{
friend class Point;
friend class IsogenyClass;
public:
int isnull() const
{return ((a1==0)&&(a2==0)&&(a3==0)&&(a4==0)&&(a6==0));}
void getai(bigint& aa1, bigint& aa2, bigint& aa3,
bigint& aa4, bigint& aa6) const
{aa1=a1; aa2=a2; aa3=a3; aa4=a4; aa6=a6; }
// input and output
void output(ostream& os) const
{ os<<"["<<a1<<","<<a2<<","<<a3<<","<<a4<<","<<a6<<"]";}
//N.B. No spaces in output for ease of input to GP
void input(istream& is);
void tex_print(ostream &) const ;
// puts out TeX-ed equation of curve; never been used
// constructors
Curve(void) // :a1(0),a2(0),a3(0),a4(0),a6(0)
{a1=0;a2=0;a3=0;a4=0;a6=0;}
Curve(const bigint& c4, const bigint& c6); //init by invariants
//check valid for elliptic curve
//if not create null curve
Curve(const bigint& aa1, const bigint& aa2, const bigint& aa3,
const bigint& aa4, const bigint& aa6)
:a1(aa1),a2(aa2),a3(aa3),a4(aa4),a6(aa6) {}
Curve(const Curve& c)
: a1(c.a1), a2(c.a2), a3(c.a3), a4(c.a4), a6(c.a6)
{}
void operator=(const Curve& c)
{ a1=c.a1; a2=c.a2; a3=c.a3; a4=c.a4; a6=c.a6; }
// no destructor is needed
// equality tests -- inherited, so you can compare Curvedata with Curve
int operator==(const Curve& f) const
{return ((a1==f.a1)&&(a2==f.a2)&&
(a3==f.a3)&&(a4==f.a4)&&(a6==f.a6));}
int operator!=(const Curve& f) const
{return ((a1!=f.a1)||(a2!=f.a2)||
(a3!=f.a3)||(a4!=f.a4)||(a6!=f.a6));}
protected:
bigint a1 ;
bigint a2 ;
bigint a3 ;
bigint a4 ;
bigint a6 ;
} ;
//
// derived class for curve with computed invariants
// NB minimalize() is a member function, flagged if ever called; by
// default not called, but see constructors for how to force.
//
class Curvedata : public Curve{
friend class CurveRed; // bug fix; ridiculous, is a derived class
friend class IsogenyClass;
public:
Curvedata() {discr_factored=0;}
Curvedata(const bigint& aa1, const bigint& aa2, const bigint& aa3,
const bigint& aa4, const bigint& aa6, int min_on_init = 0);
/*
Curvedata(const bigrational& qa1, const bigrational& qa2,
const bigrational& qa3, const bigrational& qa4,
const bigrational& qa6, bigint& scale);
*/
Curvedata(const vector<bigrational>& qai, bigint& scale);
Curvedata(const Curve& c, int min_on_init = 0);
Curvedata(const bigint& cc4, const bigint& cc6, int min_on_init = 0);
Curvedata(const Curvedata& c, int min_on_init = 0);
// nb compiler cannot generate copy because constructor from
// curve overrides.
// But default assign, destruct suffice
void operator=(const Curvedata& c);
void minimalize(); // Changes self in situ
void factor_discr()
{if(!discr_factored){ the_bad_primes=pdivs(discr); discr_factored=1; }}
Curvedata minimalize(bigint& u, bigint& r, bigint& s, bigint& t) const;
// Self unchanged and returns transformation
void transform(const bigint& r, const bigint& s, const bigint& t); // NB u = 1;
// the more general case is not implemented here
void output(ostream& os) const;
void input(istream& is);
long get_ntorsion(); // implemented in points.cc
void getbi(bigint& bb2, bigint& bb4, bigint& bb6, bigint& bb8) const
{bb2=b2; bb4=b4; bb6=b6; bb8=b8; }
void getci(bigint& cc4, bigint& cc6) const
{cc4=c4; cc6=c6; }
friend inline bigint getb2(const Curvedata& c) {return c.b2; }
friend inline bigint getb4(const Curvedata& c) {return c.b4; }
friend inline bigint getb6(const Curvedata& c) {return c.b6; }
friend inline bigint getb8(const Curvedata& c) {return c.b8; }
friend inline bigint getc4(const Curvedata& c) {return c.c4; }
friend inline bigint getc6(const Curvedata& c) {return c.c6; }
friend inline bigint getdiscr(const Curvedata& c) {return c.discr; }
friend inline int getconncomp(const Curvedata& c) {return c.conncomp; }
friend inline vector<bigint> getbad_primes(Curvedata& c)
{
if(!c.discr_factored) c.factor_discr();
return c.the_bad_primes;
}
protected:
bigint b2 ;
bigint b4 ;
bigint b6 ;
bigint b8 ;
bigint c4 ;
bigint c6 ;
bigint discr ;
int minimal_flag; // 0 if .minimalize() has not been called
int discr_factored; // 0 if discr has not yet been factored
vector<bigint> the_bad_primes; //prime divisors of discriminant
int conncomp ; // number of components (1 or 2)
long ntorsion; // 0 if .gettorsion() not called
} ;
// function to find "optimal x shift" of a given curve
Curvedata opt_x_shift(const Curvedata& C, bigint& k);
/*
//
// further derived class for curve, invariants and periods:
//
class CurvedataExtra : public Curvedata{
public:
CurvedataExtra(const Curvedata&) ;
virtual void output(ostream& os) const;
void input(istream& is)
{cout<<"*** You cannot input a CurvedataExtra -- must be just Curve\n";
exit(1); }
void getroots(bigfloat& r1, bigfloat& r2, bigfloat& r3) const
{r1=roots[0]; r2=roots[1]; r3=roots[2]; }
// NB caller should then look at conncomp to see how many are set
friend inline bigfloat getperiod(const CurvedataExtra c) {return c.period; }
protected:
bigfloat roots[3] ; // real two-division points; NB if there's only one,
// it is stored in roots[2]
bigfloat period ; // smallest real period * conncomp
} ;
*/
//
// CurveRed class call Tates algorithm as constructor,
// stores the info as member variables
// class Kodaira_code just holds an int which "codes" the type as follows:
// (this coding originally from R.G.E.Pinch)
//
// Im -> 10*m
// I*m -> 10*m+1
// I, II, III, IV -> 1, 2, 3, 4
// I*, II*. III*, IV* -> 5, 6, 7, 8
//
class Kodaira_code {
public:
int code;
//
Kodaira_code(int k = 0) : code(k) {;}
Kodaira_code(const Kodaira_code& c) : code(c.code) {;}
void operator=(int k) {code=k;}
void operator=(const Kodaira_code& c) {code=c.code;}
friend ostream& operator<<(ostream& os, const Kodaira_code& c);
};
// utility function for converting Kodaira codes to the Pari coding
// Kodaira Symbol My coding Pari Coding
// I0 0 1
// I*0 1 -1
// Im (m>0) 10*m m+4
// I*m (m>0) 10*m+1 -(m+4)
// II, III, IV 2, 3, 4 m
// II*. III*, IV* 7, 6, 5 -m
int PariKodairaCode(Kodaira_code Kod);
class Reduction_type {
public:
int ord_p_discr;
int ord_p_N;
int ord_p_j_denom;
Kodaira_code Kcode; // NB the constructor makes this from an int
int c_p;
int local_root_number;
//
Reduction_type(int opd=0, int opN=0, int opj=0, int kc=1, int cp=1, int rno=0)
: ord_p_discr(opd), ord_p_N(opN), ord_p_j_denom(opj), Kcode(kc), c_p(cp), local_root_number(rno)
{}
void operator=(const Reduction_type& r)
{ord_p_discr=r.ord_p_discr;ord_p_N=r.ord_p_N;ord_p_j_denom=r.ord_p_j_denom;
Kcode=r.Kcode;c_p=r.c_p; local_root_number=r.local_root_number;}
};
inline ostream& operator<<(ostream& os, const Reduction_type& R);
class CurveRed : public Curvedata {
friend class IsogenyClass;
protected:
map<bigint,Reduction_type> reduct_array; // maps p -> its reduction type
bigint N; //the conductor
public:
CurveRed() : Curvedata() {N=0;}
CurveRed(const Curvedata& E); // construct by Tate's algorithm
// arg E need not be minimal, but the reduced form will be
~CurveRed();
CurveRed(const CurveRed& E);
void operator=(const CurveRed& E);
// The full display function is not const, since if called they will
// compute and set the local root numbers if necessary
void display(ostream& os); // full output
void output(ostream& os) const; // just the curvedata and conductor
private:
// functions for setting local root numbers:
int neron(long p, int kod); // p = 2 or 3
void setLocalRootNumber(const bigint& p);
void setLocalRootNumber2();
void setLocalRootNumber3();
void setLocalRootNumber_not_2_or_3(const bigint& p);
public:
// member access functions:
friend inline vector<bigint> getbad_primes(const CurveRed& c)
{return c.the_bad_primes; }
friend inline bigint getconductor(const CurveRed& c) {return c.N; }
friend int getord_p_discr(const CurveRed& c, const bigint& p);
friend int getord_p_N(const CurveRed& c, const bigint& p);
friend int getord_p_j_denom(const CurveRed& c, const bigint& p);
friend int getc_p(const CurveRed& c, const bigint& p);
friend int prodcp(const CurveRed& c);
friend int LocalRootNumber(CurveRed& c, const bigint& p);
friend int GlobalRootNumber(CurveRed& c);
friend Kodaira_code getKodaira_code(const CurveRed& c, const bigint& p);
// the returned value casts as a character array; to use coded as int,
// say declared Kodaira_code Kc, just use public member Kc.code
friend bigint Trace_Frob(CurveRed& c, const bigint& p);
};
inline ostream& operator<<(ostream& os, const Curve& c)
{
c.output(os);
return os;
}
inline ostream& operator<<(ostream& os, const Curvedata& c)
{
c.output(os);
return os;
}
inline ostream& operator<<(ostream& os, const CurveRed& c)
{
c.output(os);
return os;
}
inline istream& operator>>(istream& is, Curve& c)
{
c.input(is);
return is ;
}
inline istream& operator>>(istream& is, Curvedata& c)
{
c.input(is);
return is ;
}
//Reads the curve (ai) and computes the rest.
inline int GlobalRootNumber(const Curvedata& E)
{
CurveRed C(E);
return GlobalRootNumber(C);
}
// end of file: curve.h
#endif
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