def projective_point(p):
"""
Return equivalent point with denominators removed
INPUT:
- ``P``, ``Q`` -- list/tuple of projective coordinates.
OUTPUT:
List of projective coordinates.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.constructor import projective_point
sage: projective_point([4/5, 6/5, 8/5])
[2, 3, 4]
sage: F = GF(11)
sage: projective_point([F(4), F(8), F(2)])
[4, 8, 2]
"""
from sage.rings.integer import GCD_list
from sage.arith.functions import LCM_list
try:
p_gcd = GCD_list([x.numerator() for x in p])
p_lcm = LCM_list(x.denominator() for x in p)
except AttributeError:
return p
scale = p_lcm / p_gcd
return [scale * x for x in p]
def _find_cusps(self):
r"""
Calculate the reduced representatives of the equivalence classes of
cusps for this group. Adapted from code by Ron Evans.
EXAMPLE::
sage: Gamma(8).cusps() # indirect doctest
[0, 1/4, 1/3, 3/8, 1/2, 2/3, 3/4, 1, 4/3, 3/2, 5/3, 2, 7/3, 5/2, 8/3, 3, 7/2, 11/3, 4, 14/3, 5, 6, 7, Infinity]
"""
n = self.level()
C = [QQ(x) for x in xrange(n)]
n0 = n // 2
n1 = (n + 1) // 2
for r in xrange(1, n1):
if r > 1 and gcd(r, n) == 1:
C.append(ZZ(r) / ZZ(n))
if n0 == n / 2 and gcd(r, n0) == 1:
C.append(ZZ(r) / ZZ(n0))
for s in xrange(2, n1):
for r in xrange(1, 1 + n):
if GCD_list([s, r, n]) == 1:
# GCD_list is ~40x faster than gcd, since gcd wastes loads
# of time initialising a Sequence type.
u, v = _lift_pair(r, s, n)
C.append(ZZ(u) / ZZ(v))
return [Cusp(x) for x in sorted(C)] + [Cusp(1, 0)]
def base_rays(self, fiber, points):
r"""
Return the primitive lattice vectors that generate the
direction given by the base projection of points.
INPUT:
- ``fiber`` -- a sub-lattice polytope defining the
:meth:`base_projection`.
- ``points`` -- the points to project to the base.
OUTPUT:
A tuple of primitive `\ZZ`-vectors.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: poly = LatticePolytope_PPL((-9,-6,-1,-1),(0,0,0,1),(0,0,1,0),(0,1,0,0),(1,0,0,0))
sage: fiber = next(poly.fibration_generator(2))
sage: poly.base_rays(fiber, poly.integral_points_not_interior_to_facets())
((-1, -1), (0, 1), (1, 0))
sage: p = LatticePolytope_PPL((1,0),(1,2),(-1,0))
sage: f = LatticePolytope_PPL((1,0),(-1,0))
sage: p.base_rays(f, p.integral_points())
((1),)
"""
quo = self.base_projection(fiber)
vertices = set()
for p in points:
v = quo(p).vector()
if v.is_zero():
continue
d = GCD_list(v.list())
if d > 1:
v = v.__copy__()
for i in range(v.degree()):
v[i] /= d
v.set_immutable()
vertices.add(v)
return tuple(sorted(vertices))
from sage.rings.integer import GCD_list
f = open('B-large.in', 'r')
fout = open('B-large.out', 'w')
C = int(f.readline().strip())
for c in range(C):
line = f.readline().strip().split()
N = int(line[0])
ti = [int(line[i + 1]) for i in range(N)]
ti.sort()
diffs = [ti[i + 1] - ti[i] for i in range(N - 1)]
#print diffs
#w = GCD_list(diffs)
#print w
#print (-ti[0]) % w
fout.write('Case #%d: ' % (c+1))
fout.write(str((-ti[0]) % GCD_list(diffs)))
fout.write('\n')