def discriminant(self):
"""
Return the discriminant of this ring, which is the discriminant of
the trace pairing.
.. note::
One knows that for modular abelian varieties, the
endomorphism ring should be isomorphic to an order in a
number field. However, the discriminant returned by this
function will be `2^n` ( `n =`
self.dimension()) times the discriminant of that order,
since the elements are represented as 2d x 2d
matrices. Notice, for example, that the case of a one
dimensional abelian variety, whose endomorphism ring must
be ZZ, has discriminant 2, as in the example below.
EXAMPLES::
sage: J0(33).endomorphism_ring().discriminant()
-64800
sage: J0(46).endomorphism_ring().discriminant() # long time (6s on sage.math, 2011)
24200000000
sage: J0(11).endomorphism_ring().discriminant()
2
"""
g = self.gens()
M = Matrix(ZZ,len(g), [ (g[i]*g[j]).trace()
for i in range(len(g)) for j in range(len(g)) ])
return M.determinant()
def taut_polynomial_via_smith(tri, angle, cycles = [], alpha = True, mode = "taut"):
# set up
assert tri.homology().rank() == 1 # need the polynomial ring to be a PID
ZH = group_ring(tri, angle, cycles, alpha = alpha, ring = QQ) # ditto
P = ZH.polynomial_ring()
ET = edges_to_triangles_matrix(tri, angle, cycles, ZH, P, mode = mode)
# compute via smith normal form
ETs = ET.smith_form()[0]
a = tri.countEdges()
ETs_reduced = Matrix([row[:a] for row in ETs])
return normalise_poly(ETs_reduced.determinant(), ZH, P)
def c3_func(SUK, prec=106):
r"""
Return the constant `c_3` from Smart's 1995 TCDF paper, [Sma1995]_
INPUT:
- ``SUK`` -- a group of `S`-units
- ``prec`` -- (default: 106) the precision of the real field
OUTPUT:
The constant ``c3``, as a real number
EXAMPLES::
sage: from sage.rings.number_field.S_unit_solver import c3_func
sage: K.<xi> = NumberField(x^3-3)
sage: SUK = UnitGroup(K, S=tuple(K.primes_above(3)))
sage: c3_func(SUK) # abs tol 1e-29
0.4257859134798034746197327286726
.. NOTE::
The numerator should be as close to 1 as possible, especially as the rank of the `S`-units grows large
REFERENCES:
- [Sma1995]_ p. 823
"""
R = RealField(prec)
all_places = list(SUK.primes()) + SUK.number_field().places(prec)
Possible_U = Combinations(all_places, SUK.rank())
c1 = R(0)
for U in Possible_U:
# first, build the matrix C_{i,U}
columns_of_C = []
for unit in SUK.fundamental_units():
columns_of_C.append(column_Log(SUK, unit, U, prec))
C = Matrix(SUK.rank(), SUK.rank(), columns_of_C)
# Is it invertible?
if abs(C.determinant()) > 10**(-10):
poss_c1 = C.inverse().apply_map(abs).norm(Infinity)
c1 = R(max(poss_c1, c1))
return R(0.9999999) / (c1*SUK.rank())
def taut_polynomial_via_tree_and_smith(tri, angle, cycles = [], alpha = True, mode = "taut"):
# set up
assert tri.homology().rank() == 1 # need the polynomial ring to be a PID
ZH = group_ring(tri, angle, cycles, alpha = alpha, ring = QQ) # ditto
P = ZH.polynomial_ring()
ET = edges_to_triangles_matrix(tri, angle, cycles, ZH, P, mode = mode)
_, non_tree_faces, _ = spanning_dual_tree(tri)
ET = ET.transpose()
ET = Matrix([row for i, row in enumerate(ET) if i in non_tree_faces]).transpose()
# compute via smith normal form
ETs = ET.smith_form()[0]
a = tri.countEdges()
ETs_reduced = Matrix([row[:a] for row in ETs])
return normalise_poly(ETs_reduced.determinant(), ZH, P)
