h.rk

强大的数学工具包

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                               *    rk     *
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        "rank"
        SYNTAX: X = rk(Y)
                X = rk(Y, s)
                X = rk(Y, s, b, v)


        Y is an expression whose value is an elliptic curve over Q
        or over a real quadratic number field with class number one
        or a matrix over Z, Q, polynomials over Z or Q, rational 
        functions over Q, number fields or GF(p^n) (where p is a 
        prime). In the second case, Y must be an elliptic curve
        over Q and s must be 0, 1, 2, or 3. In the third case, Y
        must be an elliptic curve over a real quadratic number field
        with class number one, s must be 0, 1, 2, b must be a single 
        precision positive number, and v must be 0 or 1.

        If Y is a matrix, X is assigned the rank of this matrix.
        
        If Y is an elliptic curve over Q, X is assigned the rank of 
        the Mordell-Weil group of Y. The method used for the computation 
        of the rank depends on the value of s:

        s = 0: This is the default case.
        If a 2-isogeny on Y over Q exists, the rank is computed
        by 2-descent using the 2-isogeny. 
        If this method is not successful or if there is no
        2-isogeny, the rank is computed by general 2-descent.
        If this method does not succeed, the rank is computed using the 
        conjecture of Birch and Swinnerton-Dyer (Manin's conditional 
        algorithm).

        s = 1:
        If a 2-isogeny on Y over Q exists, the rank is computed by 
        2-descent using the isogeny. 
        You can also use rk2d (see "?rk2d").
 
        s = 2:
        The rank is computed by general 2-descent.
        You can also use rkg2d (see "?rkg2d").
 
        s = 3:
        The rank is computed using the conjecture of Birch and 
        Swinnerton-Dyer (Manin's conditional algorithm).
        You can also use rkbsd (see "?rkbsd").

        If Y is an elliptic curve over a real quadratic number field with
        class number 1, rk tries to compute the rank of the Mordell-Weil
        group of Y 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 Y are not integers in K, rk 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.

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

        The method used for the computation of the rank depends on the
        value of s:

        s=0: This is the default case. 
        If there is a point of order 2, rk uses 2-descent via 2-isogeny.
        If Y has trivial 2-torsion, rk uses general 2-descent.
        You can also use rk2d (see "?rk2d").

        s=1: This is the same as s=0. 
        If there is a point of order 2, rk uses 2-descent via 2-isogeny.
        If Y has trivial 2-torsion, rk uses general 2-descent.
        You can also use rk2d (see "?rk2d").

        s=2: 
        The rank is computed via general 2-descent.
        You can also use rkg2d (see "?rkg2d").

        b must be a single precision positive number. It is the upper
        bound for the search for points on the homogeneous spaces 
        belonging to Y. The default value (in the case of an input rk(Y)
        or rk(Y,s)) is 20. If rk 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". 2-descent via 2-isogeny can be used over any 
	field Q(sqrt(D)) with sfp(D) < 2^30 and class number 1. 


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


        Example 2: (correct)

                rk( { { x^2+x, 2/3*z } { z+1, 3} } )


        Example 3: (correct)
                
                rk( EC( 0, 2 ), 3)       


        Example 4: (correct)
                
                rk( EC( NF(3), 0 ), 1)