h.rkg2d

强大的数学工具包

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                               *   rkg2d   *
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        "rank via general 2-descent"
        SYNTAX: r = rkg2d(E)
                r = rkg2d(E,b)
                r = rkg2d(E,b,v)

        In the first case, E is an expression whose value is an elliptic 
        curve over Q or over a real quadratic number field with class
        number 1. In the second case, E must be an elliptic curve over
        a real quadratic number field of class number 1.
        b must be a single precision positive number and v must be 0 or 1.

        If E is an elliptic curve over Q, r is assigned the rank of the 
        Mordell-Weil group of E over Q, computed via general 2-descent.

        If E is an elliptic curve over a real quadratic number field with
        class number 1, rkg2d tries to compute the rank of the Mordell-Weil
        group of E over this quadratic field K=Q(sqrt(D)). 
        You can use nfon in order to compute the rank of an elliptic curve
        over Q over the field K.

        Note: If the coefficients of E are not integers in K, rkg2d computes
        the rank of an isomorphic curve with integral coefficients. 

        Warning: If D > 100, the program does not really check whether the
        class number is 1.

        r is assigned the rank of the Mordell-Weil group. If the program 
        finds more than one possibility for the rank, r is assigned the first
        possible rank. The other possibilities can be found in the output.

        rkg2d uses general 2-descent for computing the rank.

        b must be a single precision positive number. It is the upper
        bound for the search for points on the homogeneous spaces 
        belonging to E. The default value (in the case of an input rkg2d(E))
        is 20. If rkg2d finds more than one possible rank, a second run with 
        a higher upper bound (e.g. 50 or 100) may lead to a uniquely 
        determined rank.

        v must be 0, if you want brief output, and 1, if you want detailed 
        output. The default value is 1. If you put an ";" at the end of the 
        input, v is automatically set to 0.

        Warning: General 2-descent should currently only be used over
        "small" quadratic number fields of class number 1, e.g. Q(sqrt(5)), 
        Q(sqrt(2)), Q(sqrt(13)), and Q(sqrt(3)). Otherwise the cpu times may 
        be "arbitrarily large". 


        Example 1: (correct)
                
                rkg2d( EC( 1, 2, 0, 1, 0 ) )       

        
        Example 2: (correct)
                
                rkg2d( EC( 1, 0, 0, 1, 2 ) )       


        Example 3: (correct)
                
                rkg2d( EC( NF(3), 0 ))