本文整理汇总了Python中sage.rings.all.Integer.sqrt方法的典型用法代码示例。如果您正苦于以下问题：Python Integer.sqrt方法的具体用法？Python Integer.sqrt怎么用？Python Integer.sqrt使用的例子？那么恭喜您, 这里精选的方法代码示例或许可以为您提供帮助。您也可以进一步了解该方法所在类sage.rings.all.Integer的用法示例。
在下文中一共展示了Integer.sqrt方法的1个代码示例，这些例子默认根据受欢迎程度排序。您可以为喜欢或者感觉有用的代码点赞，您的评价将有助于系统推荐出更棒的Python代码示例。
示例1: hilbert_class_polynomial
# 需要导入模块: from sage.rings.all import Integer [as 别名]
# 或者: from sage.rings.all.Integer import sqrt [as 别名]
#.........这里部分代码省略.........
AndreyTimofeev
- Sage implementation corrected by John Cremona (using corrected precision bounds from Andreas Enge)
- Magma implementation by David Kohel
EXAMPLES::
sage: hilbert_class_polynomial(-4)
x - 1728
sage: hilbert_class_polynomial(-7)
x + 3375
sage: hilbert_class_polynomial(-23)
x^3 + 3491750*x^2 - 5151296875*x + 12771880859375
sage: hilbert_class_polynomial(-37*4)
x^2 - 39660183801072000*x - 7898242515936467904000000
sage: hilbert_class_polynomial(-37*4, algorithm="magma") # optional - magma
x^2 - 39660183801072000*x - 7898242515936467904000000
sage: hilbert_class_polynomial(-163)
x + 262537412640768000
sage: hilbert_class_polynomial(-163, algorithm="sage")
x + 262537412640768000
sage: hilbert_class_polynomial(-163, algorithm="magma") # optional - magma
x + 262537412640768000
TESTS::
sage: all([hilbert_class_polynomial(d, algorithm="arb") == \
....: hilbert_class_polynomial(d, algorithm="sage") \
....: for d in range(-1,-100,-1) if d%4 in [0,1]])
True
"""
if algorithm is None:
algorithm = "arb"
D = Integer(D)
if D >= 0:
raise ValueError("D (=%s) must be negative"%D)
if not (D%4 in [0,1]):
raise ValueError("D (=%s) must be a discriminant"%D)
if algorithm == "arb":
import sage.libs.arb.arith
return sage.libs.arb.arith.hilbert_class_polynomial(D)
if algorithm == "magma":
magma.eval("R<x> := PolynomialRing(IntegerRing())")
f = str(magma.eval("HilbertClassPolynomial(%s)"%D))
return IntegerRing()['x'](f)
if algorithm != "sage":
raise ValueError("%s is not a valid algorithm"%algorithm)
from sage.quadratic_forms.binary_qf import BinaryQF_reduced_representatives
from sage.rings.all import RR, ZZ, ComplexField
from sage.functions.all import elliptic_j
# get all primitive reduced quadratic forms, (necessary to exclude
# imprimitive forms when D is not a fundamental discriminant):
rqf = BinaryQF_reduced_representatives(D, primitive_only=True)
# compute needed precision
#
# NB: [http://arxiv.org/abs/0802.0979v1], quoting Enge (2006), is
# incorrect. Enge writes (2009-04-20 email to John Cremona) "The
# source is my paper on class polynomials
# [http://hal.inria.fr/inria-00001040] It was pointed out to me by
# the referee after ANTS that the constant given there was
# wrong. The final version contains a corrected constant on p.7
# which is consistent with your example. It says:
# "The logarithm of the absolute value of the coefficient in front
# of X^j is bounded above by
#
# log (2*k_2) * h + pi * sqrt(|D|) * sum (1/A_i)
#
# independently of j", where k_2 \approx 10.163.
h = len(rqf) # class number
c1 = 3.05682737291380 # log(2*10.63)
c2 = sum([1/RR(qf[0]) for qf in rqf], RR(0))
prec = c2*RR(3.142)*RR(D).abs().sqrt() + h*c1 # bound on log
prec = prec * 1.45 # bound on log_2 (1/log(2) = 1.44..)
prec = 10 + prec.ceil() # allow for rounding error
# set appropriate precision for further computing
Dsqrt = D.sqrt(prec=prec)
R = ComplexField(prec)['t']
t = R.gen()
pol = R(1)
for qf in rqf:
a, b, c = list(qf)
tau = (b+Dsqrt)/(a<<1)
pol *= (t - elliptic_j(tau))
coeffs = [cof.real().round() for cof in pol.coefficients(sparse=False)]
return IntegerRing()['x'](coeffs)