Python Integer.class_number方法代码示例

# 需要导入模块: from sage.rings.all import Integer [as 别名]
# 或者: from sage.rings.all.Integer import class_number [as 别名]
def discriminants_with_bounded_class_number(hmax, B=None, proof=None):
    """
    Return dictionary with keys class numbers `h\le hmax` and values the
    list of all pairs `(D, f)`, with `D<0` a fundamental discriminant such
    that `Df^2` has class number `h`.  If the optional bound `B` is given,
    return only those pairs with fundamental `|D| \le B`, though `f` can
    still be arbitrarily large.

    INPUT:

    - ``hmax`` -- integer
    - `B` -- integer or None; if None returns all pairs
    - ``proof`` -- this code calls the PARI function ``qfbclassno``, so it
      could give wrong answers when ``proof``==``False``.  The default is
      whatever ``proof.number_field()`` is.  If ``proof==False`` and `B` is
      ``None``, at least the number of discriminants is correct, since it
      is double checked with Watkins's table.

    OUTPUT:

    - dictionary

    In case `B` is not given, we use Mark Watkins's: "Class numbers of
    imaginary quadratic fields" to compute a `B` that captures all `h`
    up to `hmax` (only available for `hmax\le100`).

    EXAMPLES::

        sage: v = sage.schemes.elliptic_curves.cm.discriminants_with_bounded_class_number(3)
        sage: list(v)
        [1, 2, 3]
        sage: v[1]
        [(-3, 3), (-3, 2), (-3, 1), (-4, 2), (-4, 1), (-7, 2), (-7, 1), (-8, 1), (-11, 1), (-19, 1), (-43, 1), (-67, 1), (-163, 1)]
        sage: v[2]
        [(-3, 7), (-3, 5), (-3, 4), (-4, 5), (-4, 4), (-4, 3), (-7, 4), (-8, 3), (-8, 2), (-11, 3), (-15, 2), (-15, 1), (-20, 1), (-24, 1), (-35, 1), (-40, 1), (-51, 1), (-52, 1), (-88, 1), (-91, 1), (-115, 1), (-123, 1), (-148, 1), (-187, 1), (-232, 1), (-235, 1), (-267, 1), (-403, 1), (-427, 1)]
        sage: v[3]
        [(-3, 9), (-3, 6), (-11, 2), (-19, 2), (-23, 2), (-23, 1), (-31, 2), (-31, 1), (-43, 2), (-59, 1), (-67, 2), (-83, 1), (-107, 1), (-139, 1), (-163, 2), (-211, 1), (-283, 1), (-307, 1), (-331, 1), (-379, 1), (-499, 1), (-547, 1), (-643, 1), (-883, 1), (-907, 1)]
        sage: v = sage.schemes.elliptic_curves.cm.discriminants_with_bounded_class_number(8, proof=False)
        sage: [len(v[h]) for h in v]
        [13, 29, 25, 84, 29, 101, 38, 208]

    Find all class numbers for discriminant up to 50::

        sage: sage.schemes.elliptic_curves.cm.discriminants_with_bounded_class_number(hmax=5, B=50)
        {1: [(-3, 3), (-3, 2), (-3, 1), (-4, 2), (-4, 1), (-7, 2), (-7, 1), (-8, 1), (-11, 1), (-19, 1), (-43, 1)], 2: [(-3, 7), (-3, 5), (-3, 4), (-4, 5), (-4, 4), (-4, 3), (-7, 4), (-8, 3), (-8, 2), (-11, 3), (-15, 2), (-15, 1), (-20, 1), (-24, 1), (-35, 1), (-40, 1)], 3: [(-3, 9), (-3, 6), (-11, 2), (-19, 2), (-23, 2), (-23, 1), (-31, 2), (-31, 1), (-43, 2)], 4: [(-3, 13), (-3, 11), (-3, 8), (-4, 10), (-4, 8), (-4, 7), (-4, 6), (-7, 8), (-7, 6), (-7, 3), (-8, 6), (-8, 4), (-11, 5), (-15, 4), (-19, 5), (-19, 3), (-20, 3), (-20, 2), (-24, 2), (-35, 3), (-39, 2), (-39, 1), (-40, 2), (-43, 3)], 5: [(-47, 2), (-47, 1)]}
    """
    # imports that are needed only for this function
    from sage.structure.proof.proof import get_flag
    import math
    from sage.misc.functional import round

    # deal with input defaults and type checking
    proof = get_flag(proof, 'number_field')
    hmax = Integer(hmax)

    # T stores the output
    T = {}

    # Easy case -- instead of giving error, give meaningful output
    if hmax < 1:
        return T

    if B is None:
        # Determine how far we have to go by applying Watkins's theorem.
        v = [largest_fundamental_disc_with_class_number(h) for h in range(1, hmax+1)]
        B = max([b for b,_ in v])
        fund_count = [0] + [cnt for _,cnt in v]
    else:
        # Nothing to do -- set to None so we can use this later to know not
        # to do a double check about how many we find.
        fund_count = None
        B = Integer(B)

    if B <= 2:
        # This is an easy special case, since there are no fundamental discriminants
        # this small.
        return T

    # This lower bound gets used in an inner loop below.
    from math import log
    def lb(f):
        """Lower bound on euler_phi."""
        # 1.79 > e^gamma = 1.7810724...
        if f <= 1: return 0  # don't do log(log(1)) = log(0)
        return f/(1.79*log(log(f)) + 3.0/log(log(f)))

    for D in range(-B, -2):
        D = Integer(D)
        if is_fundamental_discriminant(D):
            h_D = D.class_number(proof)
            # For each fundamental discriminant D, loop through the f's such
            # that h(D*f^2) could possibly be <= hmax.  As explained to me by Cremona,
            # we have h(D*f^2) >= (1/c)*h(D)*phi_D(f) >= (1/c)*h(D)*euler_phi(f), where
            # phi_D(f) is like euler_phi(f) but the factor (1-1/p) is replaced
            # by a factor of (1-kr(D,p)*p), where kr(D/p) is the Kronecker symbol.
            # The factor c is 1 unless D=-4 and f>1 (when c=2) or D=-3 and f>1 (when c=3).
            # Since (1-1/p) <= 1 and (1-1/p) <= (1+1/p), we see that
            #     euler_phi(f) <= phi_D(f).
            #
            # We have the following analytic lower bound on euler_phi:
#.........这里部分代码省略.........