h.ecgnp

强大的数学工具包

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                               *   ecgnp   *
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        "elliptic curve with given number of points"
        SYNTAX: X = ecgnp(p, m, D)
                X = ecgnp(m, D)

        p is an expression whose value is a prime > 3.
        In the second case, the prime p > 3 must be specified by curmod.
        m is an expression whose value is a nonnegative integer. 

        If m > 0 then m specifies the number of points on the 
        elliptic curve which is wanted.

        If m = 0 then D must not be 0 and all possible m's are
        computed and for each m the corresponding elliptic curve.
  
        D is an expression whose value is a nonpositive integer.
        If D < 0 then D specifies the imaginary quadratic number
        field Q( D^(1/2) ) where D = (p+1-m)^2 - 4*p.
        If D = 0 then D is computed in ecgnp and m must not be 0.

        X is assigned an elliptic curve with given number of points. 
        In the first case, X is an elliptic curve over Q with
        coefficients in Z/pZ and in the second case, X is an 
        elliptic curve over Z/pZ.

        If m=0, the class number of Q( D^(1/2) ) is stored in AV[0].
        The following values in AV are: AV[1]=m1, AV[2]=elliptic
        curve with m1 points over Z/pZ, and so on (see "? avfunc").

        Warning: If p > 2^30, the primality of p is not tested.
        

        Example 1: (correct)

                ecgnp(1000117, 10^6, 0) 


        Example 2: (correct)

                ecgnp(1009, 0, -35) 


        Example 3: (correct)

		curmod(1009)
                ecgnp(0, -35)


        Example 4: (incorrect)

                ecgnp(11, 0, 0)

_ERR_NR_034_ecgnp_


        Example 5: (incorrect)

                ecgnp(1009,10,0)

_ERR_NR_280