h.nffunc

强大的数学工具包

                                * * * * * * *
                                *   nffunc  *
                                * * * * * * *

        
        If you need help in entering algebraic numbers, polynomials
        over number fields, matrices and vectors over these structures
        or elliptic curves and points of elliptic curves over a number
        field, please type "? NF".

        For additional information on using functions over number 
        fields, please type "? curnf".

        Structures over number fields, which may be considered as
        structures over Q, are usually treated as structures over
        the rationals. If you want to lift them to the number field,
        specified by curnf, you can use the switch nfon / nfoff.
        After nfon in the following functions - cond, ecinf, fact, rk,
        rk2d, rkg2d, sfp, taalg, tors - the algorithms for structures 
        over number fields are used for the structures over Q. This is 
        valid until nfoff.

        The following functions are available over number fields:

        +           aftopr      discec      mod         swnf       
        -           bitrans     discnf      nftopol     taalg    
        *           chcoef      dmconstr    norm        tavb2  
        :           chpol       eccoef      prtoaf      tavb4        
        ^           classnum    ecinf       ptcoef      tavb6         
        /           coef        elsubst     puto        tavb8 
        ==          cond        fact        relcln      tavc4
        !=          conjug      gcd         rk          tavc6
        &&          curec       intbas      rk2d        tors          
        ||          declaw      isptec      rkg2d       trace 
                    deg         jinv        scalar      transp
                    deriv       linequ      sfp         unit 
                    det         mcoef       sort        vcoef


        '+'             means   addition;

        '-'                     subtraction;

        '*'                     multiplication;

        ':'                     quotient;

        '^'                     exponentiation;

        '/'                     division;

        '=='                    equal;

        '!='                    not equal;

        '&&'                    and;

        '||'                    or;

        'aftopr'                conversion from affine to projective 
                                representation;

        'bitrans'               birational transformation of an elliptic
                                curve or a point;

        'chcoef'                change coefficient of a polynomial;

        'chpol'                 characteristic polynomial;

        'classnum'              class num;

        'coef'                  coefficient of a polynomial;

        'cond'                  conductor;

        'conjug'                conjugate;

        'curec'                 current elliptic curve;

        'declaw'                decomposition law;

        'deg'                   degree;

        'deriv'                 derivation;

        'det'                   determinant;

        'discec'                discriminant of an elliptic curve;

        'discnf'                discriminant of a number field;

        'dmconstr'              diagonal matrix construction;

        'eccoef'                elliptic curve coefficient;

        'ecinf'                 elliptic curve information;

        'elsubst'               element substitution in matrices or vectors;

        'fact'                  factorization;

        'gcd'                   greatest common divisor;

        'intbas'                integral basis;

        'isptec'                is point on an elliptic curve ? ;

        'jinv'                  j-invariant;

        'linequ'                solution of a system of linear equations;

        'mcoef'                 matrix coefficient;

        'mod'                   module homomorphism;

        'nftopol'               number field to polynomial;

        'norm'                  norm of an algebraic number;

        'prtoaf'                conversion from projective to affine 
                                representation;

        'ptcoef'                point on elliptic curve coefficient;

        'puto'                  display internal representaion;

        'relcln'                relative class number;

        'rk'                    rank;

        'rk2d'                  rank via 2-descent;

        'rkg2d'                 rank via general 2-descent;

        'scalar'                euclidian scalar product;

        'sfp'                   squarefree part;

        'sort'                  sort polynomial;
        
        'swnf'                  short Weierstrass normal form;

        'taalg'                 Tate's algorithm;

        'tavb2'                 Tate's value b2;

        'tavb4'                 Tate's value b4;

        'tavb6'                 Tate's value b6;

        'tavb8'                 Tate's value b8;

        'tavc4'                 Tate's value c4;

        'tavc6'                 Tate's value c6;

        'tors'                  torsion group;

        'trace'                 trace of an algebraic number;

        'transp'                transpose;

        'unit'                  unit group;

        'vcoef'                 vector coefficient.