h.rk2d

来自「强大的数学工具包」· RK2D 代码 · 共 68 行

                               * * * * * * *
                               *   rk2d    *
                               * * * * * * *

        "rank via 2-descent"
        SYNTAX: r = rk2d(E)
                r = rk2d(E,b)
                r = rk2d(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, the order of the torsion group of 
        E must be even. r is assigned the rank of the Mordell-Weil group of 
        E over Q, computed using 2-descent via 2-isogeny.

        If E is an elliptic curve over a real quadratic number field with
        class number 1, rk2d 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, rk2d 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.

        In the number field case, if there is a point of order 2, rk2d uses 
        2-descent via 2-isogeny. If E has trivial 2-torsion, rk2d uses general 
        2-descent.

        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 rk2d(E))
        is 20. If rk2d 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.

        
        Example 1: (correct)
                
                rk2d( EC( 1, 2, 0, 1, 0 ) )       

        
        Example 2: (incorrect)
                
                rk2d( EC( 1, 0, 0, 1, 2 ) )       

_ERR_NR_149

        
        Example 3: (correct)
                
                rk2d( EC( NF(3), 0 ))