h.ecinf

强大的数学工具包

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                               *   ecinf   *
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        "elliptic curve information"
        SYNTAX: X = ecinf(E)

        E is an expression whose value is an elliptic curve over
        Q, Z/pZ (where p is a prime), GF(2^n) or over a number field.

        If E is an elliptic curve over Q, X is assigned a global 
        minimal model of E; otherwise, X is assigned E.

        You can use nfon in order to consider the elliptic curve over Q 
        over the quadratic number field, specified by curnf. 
        
        ecinf displays:
                - Tate's values b2, b4, b6, b8;
                - Tate's values c4, c6;
                - the discriminant (and its prime ideal factorization);
                - the j-invariant [and the factorization of its 
                  denominator];
                - [a global minimal model and the parameters of 
                  the transformation];
                - (the conductor and its prime ideal factorization);
                - (the Kodaira and Neron symbols for the primes
                  of bad reduction).
        Expressions in parentheses [] are only computed for elliptic
        curves over Q.
        Expressions in parentheses () are only computed for elliptic
        curves over Q or quadratic number fields.

        If the computation is done over Z/pZ, GF(2^n) or number fields,
        AV[0]=b2, AV[1]=b4, AV[2]=b6, AV[3]=b8, AV[4]=c4, AV[5]=c6,
        AV[6]=discriminant, AV[7]=j-invariant. 
        If the computation is done over Q, the first 7 values in AV
        are the same as above. After AV[6] is the factorization of
        the discriminant is stored in the form factor - exponent in
        increasing order. After that the j-invariant with the same
        form of factorization of its denominator is stored in AV.
        Afterwards the parameters of the transformation to the minimal
        model are stored in the order r,s,t,u in AV. At least the
        conductor with its factorization in the above form is stored
        in AV (see "?avfunc").

        For the representation of ideals in quadratic number fields,
        please see "?fact".


        Example 1: (correct)

                ecinf(EC(3, 9, 27, 81, 3^6))


        Example 2: (correct)

                ecinf(EC(NF(A), 9))


        Example 3: (correct)

                ecinf(EC(1/7, 2))