SharedPierre Vanhove.sagewsOpen in CoCalc

def orthogonal_sublattice(self, *args, **kwds):
r"""
The sublattice (in the dual lattice) orthogonal to the
sublattice spanned by the cone.

Let M= self.dual_lattice() be the lattice dual to the
ambient lattice of the given cone \sigma. Then, in the
notation of [Fu1993]_, this method returns the sublattice

.. MATH::

M(\sigma) \stackrel{\text{def}}{=}
\sigma^\perp \cap M
\subset M

INPUT:

- either nothing or something that can be turned into an element of
this lattice.

OUTPUT:

- if no arguments were given, a :class:toric sublattice
<sage.geometry.toric_lattice.ToricLattice_sublattice_with_basis>,
otherwise the corresponding element of it.

EXAMPLES::

sage: c = Cone([(1,1,1), (1,-1,1), (-1,-1,1), (-1,1,1)])
sage: c.orthogonal_sublattice()
Sublattice <>
sage: c12 = Cone([(1,1,1), (1,-1,1)])
sage: c12.sublattice()
Sublattice <N(1, -1, 1), N(0, 1, 0)>
sage: c12.orthogonal_sublattice()
Sublattice <M(1, 0, -1)>
"""
if "_orthogonal_sublattice" not in self.__dict__:
try:
self._orthogonal_sublattice = self.sublattice_quotient().dual()
except AttributeError:
N = self.lattice()
n = N.dimension()
basis = self.rays().basis()
r = len(basis)
Nsigma = column_matrix(ZZ, r, n, [N.coordinates(v) for v in basis])
D, U, V = Nsigma.smith_form()  # D = U*N*V <=> N = Uinv*D*Vinv

M = self.dual_lattice()
# basis for the dual spanned lattice
basis = [U.row(i) for i in range(r, n)]
self._orthogonal_sublattice = M.submodule_with_basis(basis)
if args or kwds:
return self._orthogonal_sublattice(*args, **kwds)
else:
return self._orthogonal_sublattice

sage.geometry.cone.ConvexRationalPolyhedralCone.orthogonal_sublattice = orthogonal_sublattice

R.<x,y>=QQ[]
Asunset = lambda m1, m2, m3, p2: (m1^2*x+m2^2*y+m3^2)*(x*y+x+y)-p2*x*y
Psunset = Asunset(1,2,3,4).newton_polytope()
Psunset

A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 6 vertices
Delta = LatticePolytope(Psunset.vertices_list())
assert len(Delta.interior_points()) == 1
center = Delta.interior_points()[0]
Delta = LatticePolytope([p - center for p in Delta.vertices()])
assert Delta.is_reflexive()
Delta.plot3d()

3D rendering not yet implemented
DeltaP = Delta.polar()

DeltaP.points()

N( 1, 0), N( 1, 1), N( 0, -1), N( 0, 1), N(-1, 0), N(-1, -1), N( 0, 0) in 2-d lattice N
m = matrix(ZZ,[[1,0],[1,1],[0,-1],[0,1],[-1,0],[-1,0],[-1,-1],[0,0]])

p = LatticePolytope(m)

p;

2-d reflexive polytope #9 in 2-d lattice M


p.normal_form()

M( 1, 0), M( 0, 1), M( 1, -1), M(-1, 1), M( 0, -1), M(-1, 0) in 2-d lattice M








7bb972ec-6c4b-4e98-96fa-345913a7884e︠



b0285a94-4d98-4d94-bc4c-f955283863c1︠





DeltaP.is_reflexive()
DeltaP.plot3d()

True
3D rendering not yet implemented









































X = CPRFanoToricVariety(Delta)
Mc = X.Mori_cone()
Mc

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1043, in execute exec compile(block+'\n', '', 'single', flags=compile_flags) in namespace, locals File "", line 1, in <module> TypeError: 'LatticePolytopeClass' object is not callable
Mc.rays()

(-1, 1, 1, 0, 0, 0, -1), ( 0, 0, 0, 1, -1, 1, -1), ( 0, 0, 1, 0, 1, -1, -1), ( 0, 1, 0, -1, 1, 0, -1), ( 1, -1, 0, 1, 0, 0, -1), ( 1, 0, -1, 0, 0, 1, -1) in Ambient free module of rank 7 over the principal ideal domain Integer Ring
X.Mori_cone?

/ext/sage/sage-8.0/local/lib/python2.7/site-packages/urllib3/contrib/pyopenssl.py:46: DeprecationWarning: OpenSSL.rand is deprecated - you should use os.urandom instead import OpenSSL.SSL
File: /ext/sage/sage-8.0/src/sage/schemes/toric/variety.py
Signature : X.Mori_cone(self)
Docstring :
Returns the Mori cone of "self".

OUTPUT:

* "cone".

Note:

* The Mori cone is dual to the Kähler cone.

* We think of the Mori cone as living inside the row span of
the Gale transform matrix (computed by
"self.fan().Gale_transform()").

* The points in the Mori cone are the effective curves in the
variety.

* The "i"-th entry in each Mori vector is the intersection
number of the curve corresponding to the generator of the
"i"-th ray of the fan with the corresponding divisor class. The
very last entry is associated to the origin of the fan lattice.

* The Mori vectors are also known as the gauged linear sigma
model charge vectors.

EXAMPLES:

sage: P4_11169 = toric_varieties.P4_11169_resolved()
sage: P4_11169.Mori_cone()
2-d cone in 7-d lattice
sage: P4_11169.Mori_cone().rays()
(3, 2, 0, 0, 0,  1, -6),
(0, 0, 1, 1, 1, -3,  0)
in Ambient free module of rank 7
over the principal ideal domain Integer Ring



R.<x1,x2,x3> = QQ[]
A3 = lambda m1,m2,m3,m4,s: -s*(m1^2*x1+m2^2*x2+m3^2*x3+m4^2)*(x1*x2*x3+x1*x2+x1*x3+x2*x3)+x1*x2*x3
PA3 = A3(1,2,5,9,9).newton_polytope()
PA3

A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 12 vertices
Delta = LatticePolytope(PA3.vertices_list())
assert len(Delta.interior_points()) == 1
center = Delta.interior_points()[0]
Delta = LatticePolytope([p - center for p in Delta.vertices()])
assert Delta.is_reflexive()
Delta.plot3d()

3D rendering not yet implemented
from fibration import *
from elliptic_fibration import *

slices = reflexive_slices(Delta, symmetries=True)
len(slices)

2
slices = [slice_polytope(Delta, normal=slice.normal) for slice in slices]
slices

[2-dimensional reflexive slice of a 3-dimensional polytope. Points: [3, 4, 5, 7, 8, 9, 12] Skeleton points: [3, 4, 5, 7, 8, 9] Normal to the slice hyperplane: (1, 0, 0) Index of the slice polytope: 9, 2-dimensional reflexive slice of a 3-dimensional polytope. Points: [1, 3, 9, 11, 12] Skeleton points: [1, 3, 9, 11] Normal to the slice hyperplane: (1, 0, 1) Index of the slice polytope: 3]
DeltaPol = Delta.polar()

slicesPol = reflexive_slices(DeltaPol, symmetries=True)
len(slices)

2
Delta.index()

1529
DeltaPol.index()

2355
DeltaPol.plot3d()

3D rendering not yet implemented




1eb3e76b-aecc-4960-8d21-d9d027597d44︠



DeltaPol.points()

N(-1, 0, -1), N(-1, -1, 0), N( 1, 1, 1), N( 1, 1, 0), N( 1, 0, 0), N( 1, 0, 1), N( 0, 0, -1), N( 0, 0, 1), N( 0, 1, 0), N( 0, -1, 0), N( 0, -1, -1), N( 0, 1, 1), N(-1, 0, 0), N(-1, -1, -1), N( 0, 0, 0) in 3-d lattice N




slicesPol

(2-dimensional reflexive slice of a 3-dimensional polytope. Skeleton points: [1, 2, 3, 6, 7, 13] Normal to the slice hyperplane: (1, -1, 0) Index of the slice polytope: 9, 2-dimensional reflexive slice of a 3-dimensional polytope. Skeleton points: [0, 1, 3, 5] Normal to the slice hyperplane: (1, -1, -1) Index of the slice polytope: 3)
























slicesPol = [slice_polytope(DeltaPol, normal=slice.normal) for slice in slices]
slicesPol

[2-dimensional reflexive slice of a 3-dimensional polytope. Points: [6, 7, 8, 9, 10, 11, 14] Skeleton points: [6, 7, 8, 9, 10, 11] Normal to the slice hyperplane: (1, 0, 0) Index of the slice polytope: 9, [8, 9, 14]]
X = CPRFanoToricVariety(Delta)
Mc = X.Mori_cone()
Mc

11-d cone in 15-d lattice
Mc.rays()

( 0, 1, -1, 2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1), ( 0, 0, 1, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1), ( 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0), ( 0, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0), ( 1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0), (-1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0), ( 1, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0), (-1, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0), ( 1, -1, 1, -1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0), (-1, 1, -1, 1, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0), ( 1, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0), (-1, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0), ( 1, 0, 0, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0), (-1, 0, 0, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0), ( 0, 1, 0, 1, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0), ( 0, -1, 0, -1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0), ( 1, -1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0), (-1, 1, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0), ( 1, -1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (-1, 1, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) in Ambient free module of rank 15 over the principal ideal domain Integer Ring
C = Cone([(1,0)], lattice=ZZ^2)
C

1-d cone in 2-d lattice
C.dual()

2-d cone in 2-d lattice
#Introduced by https://trac.sagemath.org/attachment/ticket/13183/trac_13183_index_of_codomain_cone.patch

Delta

3-d reflexive polytope in 3-d lattice M
Delta.vertices()

M(-1, 0, 0), M(-1, 0, 1), M(-1, 1, 0), M( 0, -1, 0), M( 0, -1, 1), M( 0, 0, -1), M( 1, 0, 0), M( 0, 0, 1), M( 0, 1, -1), M( 0, 1, 0), M( 1, -1, 0), M( 1, 0, -1) in 3-d lattice M
Delta_p = Delta.polar()
Delta_p.vertices()

N(-1, 0, -1), N(-1, -1, 0), N( 1, 1, 1), N( 1, 1, 0), N( 1, 0, 0), N( 1, 0, 1), N( 0, 0, -1), N( 0, 0, 1), N( 0, 1, 0), N( 0, -1, 0), N( 0, -1, -1), N( 0, 1, 1), N(-1, 0, 0), N(-1, -1, -1) in 3-d lattice N
Delta.index(), Delta_p.index()

(1529, 2355)
load('fibrations.py')

%time
slices = reflexive_slices(Delta_p)
len(slices)

2 CPU time: 0.86 s, Wall time: 5.37 s
for slice in slices:
fib = Fibration(Delta_p, slice.normal, True)
fib.show()
show(fib.slice().top, fib.slice().bottom)

A fibration of:
$\displaystyle \Delta^{3}_{2355}$
Corresponding to:
----------------------------------------------------------------------
2-dimensional reflexive slice of a 3-dimensional polytope. Points: [1, 2, 3, 6, 7, 13, 14] Skeleton points: [1, 2, 3, 6, 7, 13] Normal to the slice hyperplane: (1, -1, 0) Vertices of the slice polytope: M(-1, -1), M(-1, 0), M( 0, -1), M( 1, 1), M( 0, 1), M( 1, 0) in 2-d lattice M
----------------------------------------------------------------------
Coordinate points:
$\displaystyle \left(\begin{array}{rrrrrrrrrrrrrr} -1 & -1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 \\ 0 & -1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 & 0 & 1 & -1 & 1 & 0 & 0 & -1 & 1 & 0 & -1 \end{array}\right)$
Monomial points:
$\displaystyle \left(\begin{array}{rrrrrrrrrrrrr} -1 & -1 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & -1 & -1 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & -1 & 0 & 1 & -1 & 0 & 0 & -1 & 0 \end{array}\right)$
Equation of a hypersurface:
$\displaystyle a_{6} z_{2}^{2} z_{3}^{2} z_{4}^{2} z_{5}^{2} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} + a_{9} z_{0} z_{2}^{2} z_{3}^{2} z_{4} z_{5} z_{6} z_{7} z_{8}^{2} z_{11}^{2} z_{12} + a_{7} z_{1} z_{2}^{2} z_{3} z_{4} z_{5}^{2} z_{7}^{2} z_{8} z_{9} z_{11}^{2} z_{12} + a_{11} z_{0} z_{2} z_{3}^{2} z_{4}^{2} z_{5} z_{6}^{2} z_{8} z_{9} z_{10}^{2} z_{13} + a_{10} z_{1} z_{2} z_{3} z_{4}^{2} z_{5}^{2} z_{6} z_{7} z_{9}^{2} z_{10}^{2} z_{13} + a_{8} z_{0}^{2} z_{2} z_{3}^{2} z_{4} z_{6}^{2} z_{8}^{2} z_{10} z_{11} z_{12} z_{13} + a_{12} z_{0} z_{1} z_{2} z_{3} z_{4} z_{5} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} z_{12} z_{13} + a_{4} z_{1}^{2} z_{2} z_{4} z_{5}^{2} z_{7}^{2} z_{9}^{2} z_{10} z_{11} z_{12} z_{13} + a_{2} z_{0}^{2} z_{1} z_{2} z_{3} z_{6} z_{7} z_{8}^{2} z_{11}^{2} z_{12}^{2} z_{13} + a_{1} z_{0} z_{1}^{2} z_{2} z_{5} z_{7}^{2} z_{8} z_{9} z_{11}^{2} z_{12}^{2} z_{13} + a_{5} z_{0}^{2} z_{1} z_{3} z_{4} z_{6}^{2} z_{8} z_{9} z_{10}^{2} z_{12} z_{13}^{2} + a_{3} z_{0} z_{1}^{2} z_{4} z_{5} z_{6} z_{7} z_{9}^{2} z_{10}^{2} z_{12} z_{13}^{2} + a_{0} z_{0}^{2} z_{1}^{2} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} z_{12}^{2} z_{13}^{2}$
Projection to the base:
($\displaystyle z_{4} z_{5} z_{9} z_{10}$, $\displaystyle z_{0} z_{8} z_{11} z_{12}$)
Coordinate on the base: z4
Fiber over (t^(1),1):
$\displaystyle (a_{6} t^{2} + a_{9} t) z_{2}^{2} z_{3}^{2} z_{6} z_{7} + (a_{7} t) z_{1} z_{2}^{2} z_{3} z_{7}^{2} + (a_{11} t^{2} + a_{8} t) z_{2} z_{3}^{2} z_{6}^{2} z_{13} + (a_{10} t^{2} + a_{12} t + a_{2}) z_{1} z_{2} z_{3} z_{6} z_{7} z_{13} + (a_{4} t + a_{1}) z_{1}^{2} z_{2} z_{7}^{2} z_{13} + (a_{5} t) z_{1} z_{3} z_{6}^{2} z_{13}^{2} + (a_{3} t + a_{0}) z_{1}^{2} z_{6} z_{7} z_{13}^{2}$
d3-based renderer not yet implemented
d3-based renderer not yet implemented
A fibration of:
$\displaystyle \Delta^{3}_{2355}$
Corresponding to:
----------------------------------------------------------------------
2-dimensional reflexive slice of a 3-dimensional polytope. Points: [0, 1, 3, 5, 14] Skeleton points: [0, 1, 3, 5] Normal to the slice hyperplane: (1, -1, -1) Vertices of the slice polytope: M(-1, -1), M(-1, 0), M( 1, 1), M( 1, 0) in 2-d lattice M
----------------------------------------------------------------------
Coordinate points:
$\displaystyle \left(\begin{array}{rrrrrrrrrrrrrr} -1 & -1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 \\ 0 & -1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 & 0 & 1 & -1 & 1 & 0 & 0 & -1 & 1 & 0 & -1 \end{array}\right)$
Monomial points:
$\displaystyle \left(\begin{array}{rrrrrrrrrrrrr} -1 & -1 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & -1 & -1 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & -1 & 0 & 1 & -1 & 0 & 0 & -1 & 0 \end{array}\right)$
Equation of a hypersurface:
$\displaystyle a_{6} z_{2}^{2} z_{3}^{2} z_{4}^{2} z_{5}^{2} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} + a_{9} z_{0} z_{2}^{2} z_{3}^{2} z_{4} z_{5} z_{6} z_{7} z_{8}^{2} z_{11}^{2} z_{12} + a_{7} z_{1} z_{2}^{2} z_{3} z_{4} z_{5}^{2} z_{7}^{2} z_{8} z_{9} z_{11}^{2} z_{12} + a_{11} z_{0} z_{2} z_{3}^{2} z_{4}^{2} z_{5} z_{6}^{2} z_{8} z_{9} z_{10}^{2} z_{13} + a_{10} z_{1} z_{2} z_{3} z_{4}^{2} z_{5}^{2} z_{6} z_{7} z_{9}^{2} z_{10}^{2} z_{13} + a_{8} z_{0}^{2} z_{2} z_{3}^{2} z_{4} z_{6}^{2} z_{8}^{2} z_{10} z_{11} z_{12} z_{13} + a_{12} z_{0} z_{1} z_{2} z_{3} z_{4} z_{5} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} z_{12} z_{13} + a_{4} z_{1}^{2} z_{2} z_{4} z_{5}^{2} z_{7}^{2} z_{9}^{2} z_{10} z_{11} z_{12} z_{13} + a_{2} z_{0}^{2} z_{1} z_{2} z_{3} z_{6} z_{7} z_{8}^{2} z_{11}^{2} z_{12}^{2} z_{13} + a_{1} z_{0} z_{1}^{2} z_{2} z_{5} z_{7}^{2} z_{8} z_{9} z_{11}^{2} z_{12}^{2} z_{13} + a_{5} z_{0}^{2} z_{1} z_{3} z_{4} z_{6}^{2} z_{8} z_{9} z_{10}^{2} z_{12} z_{13}^{2} + a_{3} z_{0} z_{1}^{2} z_{4} z_{5} z_{6} z_{7} z_{9}^{2} z_{10}^{2} z_{12} z_{13}^{2} + a_{0} z_{0}^{2} z_{1}^{2} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} z_{12}^{2} z_{13}^{2}$
Projection to the base:
($\displaystyle z_{4} z_{6} z_{9} z_{10}^{2} z_{13}$, $\displaystyle z_{2} z_{7} z_{8} z_{11}^{2} z_{12}$)
Coordinate on the base: z4
Fiber over (t^(1),1):
$\displaystyle a_{0} z_{0}^{2} z_{1}^{2} + (a_{5} t + a_{2}) z_{0}^{2} z_{1} z_{3} + (a_{8} t) z_{0}^{2} z_{3}^{2} + (a_{3} t + a_{1}) z_{0} z_{1}^{2} z_{5} + (a_{12} t) z_{0} z_{1} z_{3} z_{5} + (a_{11} t^{2} + a_{9} t) z_{0} z_{3}^{2} z_{5} + (a_{4} t) z_{1}^{2} z_{5}^{2} + (a_{10} t^{2} + a_{7} t) z_{1} z_{3} z_{5}^{2} + (a_{6} t^{2}) z_{3}^{2} z_{5}^{2}$
d3-based renderer not yet implemented
d3-based renderer not yet implemented
load("elliptic_fibration.sage")

slice = slices[0]
fib = EllipticFibration(Delta_p, slice.normal)
#fib.predict_singular_fibers()
fib.show()

A fibration of:
$\displaystyle \Delta^{3}_{2355}$
Corresponding to:
----------------------------------------------------------------------
2-dimensional reflexive slice of a 3-dimensional polytope. Points: [1, 2, 3, 6, 7, 13, 14] Skeleton points: [1, 2, 3, 6, 7, 13] Normal to the slice hyperplane: (1, -1, 0) Vertices of the slice polytope: M(-1, -1), M(-1, 0), M( 0, -1), M( 1, 1), M( 0, 1), M( 1, 0) in 2-d lattice M
----------------------------------------------------------------------
Coordinate points:
$\displaystyle \left(\begin{array}{rrrrrrrrrrrrrr} -1 & -1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 \\ 0 & -1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 & 0 & 1 & -1 & 1 & 0 & 0 & -1 & 1 & 0 & -1 \end{array}\right)$
Monomial points:
$\displaystyle \left(\begin{array}{rrrrrrrrrrrrr} -1 & -1 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & -1 & -1 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & -1 & 0 & 1 & -1 & 0 & 0 & -1 & 0 \end{array}\right)$
Equation of a hypersurface:
$\displaystyle a_{6} z_{2}^{2} z_{3}^{2} z_{4}^{2} z_{5}^{2} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} + a_{9} z_{0} z_{2}^{2} z_{3}^{2} z_{4} z_{5} z_{6} z_{7} z_{8}^{2} z_{11}^{2} z_{12} + a_{7} z_{1} z_{2}^{2} z_{3} z_{4} z_{5}^{2} z_{7}^{2} z_{8} z_{9} z_{11}^{2} z_{12} + a_{11} z_{0} z_{2} z_{3}^{2} z_{4}^{2} z_{5} z_{6}^{2} z_{8} z_{9} z_{10}^{2} z_{13} + a_{10} z_{1} z_{2} z_{3} z_{4}^{2} z_{5}^{2} z_{6} z_{7} z_{9}^{2} z_{10}^{2} z_{13} + a_{8} z_{0}^{2} z_{2} z_{3}^{2} z_{4} z_{6}^{2} z_{8}^{2} z_{10} z_{11} z_{12} z_{13} + a_{12} z_{0} z_{1} z_{2} z_{3} z_{4} z_{5} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} z_{12} z_{13} + a_{4} z_{1}^{2} z_{2} z_{4} z_{5}^{2} z_{7}^{2} z_{9}^{2} z_{10} z_{11} z_{12} z_{13} + a_{2} z_{0}^{2} z_{1} z_{2} z_{3} z_{6} z_{7} z_{8}^{2} z_{11}^{2} z_{12}^{2} z_{13} + a_{1} z_{0} z_{1}^{2} z_{2} z_{5} z_{7}^{2} z_{8} z_{9} z_{11}^{2} z_{12}^{2} z_{13} + a_{5} z_{0}^{2} z_{1} z_{3} z_{4} z_{6}^{2} z_{8} z_{9} z_{10}^{2} z_{12} z_{13}^{2} + a_{3} z_{0} z_{1}^{2} z_{4} z_{5} z_{6} z_{7} z_{9}^{2} z_{10}^{2} z_{12} z_{13}^{2} + a_{0} z_{0}^{2} z_{1}^{2} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} z_{12}^{2} z_{13}^{2}$
Projection to the base:
($\displaystyle z_{4} z_{5} z_{9} z_{10}$, $\displaystyle z_{0} z_{8} z_{11} z_{12}$)
Coordinate on the base: z4
Fiber over (t^(1),1):
$\displaystyle (a_{6} t^{2} + a_{9} t) z_{2}^{2} z_{3}^{2} z_{6} z_{7} + (a_{7} t) z_{1} z_{2}^{2} z_{3} z_{7}^{2} + (a_{11} t^{2} + a_{8} t) z_{2} z_{3}^{2} z_{6}^{2} z_{13} + (a_{10} t^{2} + a_{12} t + a_{2}) z_{1} z_{2} z_{3} z_{6} z_{7} z_{13} + (a_{4} t + a_{1}) z_{1}^{2} z_{2} z_{7}^{2} z_{13} + (a_{5} t) z_{1} z_{3} z_{6}^{2} z_{13}^{2} + (a_{3} t + a_{0}) z_{1}^{2} z_{6} z_{7} z_{13}^{2}$
Top: ExtA3. Bottom: ExtA3.
F_0: I_4. F_infinity: I_4.
slice = slices[1]
fib = EllipticFibration(Delta_p, slice.normal)
fib.predict_singular_fibers()
fib.show()

[I_1, I_1, I_1, I_1, I_1, I_1, I_1, I_1, I_1, I_1, I_1, I_1, I*_0, I*_0]
A fibration of:
$\displaystyle \Delta^{3}_{2355}$
Corresponding to:
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2-dimensional reflexive slice of a 3-dimensional polytope. Points: [0, 1, 3, 5, 14] Skeleton points: [0, 1, 3, 5] Normal to the slice hyperplane: (1, -1, -1) Vertices of the slice polytope: M(-1, -1), M(-1, 0), M( 1, 1), M( 1, 0) in 2-d lattice M
----------------------------------------------------------------------
Coordinate points:
$\displaystyle \left(\begin{array}{rrrrrrrrrrrrrr} -1 & -1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 \\ 0 & -1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 & 0 & 1 & -1 & 1 & 0 & 0 & -1 & 1 & 0 & -1 \end{array}\right)$
Monomial points:
$\displaystyle \left(\begin{array}{rrrrrrrrrrrrr} -1 & -1 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & -1 & -1 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & -1 & 0 & 1 & -1 & 0 & 0 & -1 & 0 \end{array}\right)$
Equation of a hypersurface:
$\displaystyle a_{6} z_{2}^{2} z_{3}^{2} z_{4}^{2} z_{5}^{2} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} + a_{9} z_{0} z_{2}^{2} z_{3}^{2} z_{4} z_{5} z_{6} z_{7} z_{8}^{2} z_{11}^{2} z_{12} + a_{7} z_{1} z_{2}^{2} z_{3} z_{4} z_{5}^{2} z_{7}^{2} z_{8} z_{9} z_{11}^{2} z_{12} + a_{11} z_{0} z_{2} z_{3}^{2} z_{4}^{2} z_{5} z_{6}^{2} z_{8} z_{9} z_{10}^{2} z_{13} + a_{10} z_{1} z_{2} z_{3} z_{4}^{2} z_{5}^{2} z_{6} z_{7} z_{9}^{2} z_{10}^{2} z_{13} + a_{8} z_{0}^{2} z_{2} z_{3}^{2} z_{4} z_{6}^{2} z_{8}^{2} z_{10} z_{11} z_{12} z_{13} + a_{12} z_{0} z_{1} z_{2} z_{3} z_{4} z_{5} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} z_{12} z_{13} + a_{4} z_{1}^{2} z_{2} z_{4} z_{5}^{2} z_{7}^{2} z_{9}^{2} z_{10} z_{11} z_{12} z_{13} + a_{2} z_{0}^{2} z_{1} z_{2} z_{3} z_{6} z_{7} z_{8}^{2} z_{11}^{2} z_{12}^{2} z_{13} + a_{1} z_{0} z_{1}^{2} z_{2} z_{5} z_{7}^{2} z_{8} z_{9} z_{11}^{2} z_{12}^{2} z_{13} + a_{5} z_{0}^{2} z_{1} z_{3} z_{4} z_{6}^{2} z_{8} z_{9} z_{10}^{2} z_{12} z_{13}^{2} + a_{3} z_{0} z_{1}^{2} z_{4} z_{5} z_{6} z_{7} z_{9}^{2} z_{10}^{2} z_{12} z_{13}^{2} + a_{0} z_{0}^{2} z_{1}^{2} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} z_{12}^{2} z_{13}^{2}$
Moduli parameters:
[$\displaystyle \frac{a_{1} a_{11}}{a_{12}^{2}}$, $\displaystyle \frac{a_{2} a_{10}}{a_{12}^{2}}$, $\displaystyle \frac{a_{3} a_{12}}{a_{0} a_{10}}$, $\displaystyle \frac{a_{4} a_{11}}{a_{10} a_{12}}$, $\displaystyle \frac{a_{5} a_{12}}{a_{0} a_{11}}$, $\displaystyle \frac{a_{0} a_{6}}{a_{12}^{2}}$, $\displaystyle \frac{a_{0} a_{7} a_{11}}{a_{12}^{3}}$, $\displaystyle \frac{a_{8} a_{10}}{a_{11} a_{12}}$, $\displaystyle \frac{a_{0} a_{9} a_{10}}{a_{12}^{3}}$]
Projection to the base:
($\displaystyle z_{4} z_{6} z_{9} z_{10}^{2} z_{13}$, $\displaystyle z_{2} z_{7} z_{8} z_{11}^{2} z_{12}$)
Coordinate on the base: z4
Fiber over (t^(1),1):
$\displaystyle a_{0} z_{0}^{2} z_{1}^{2} + (a_{5} t + a_{2}) z_{0}^{2} z_{1} z_{3} + (a_{8} t) z_{0}^{2} z_{3}^{2} + (a_{3} t + a_{1}) z_{0} z_{1}^{2} z_{5} + (a_{12} t) z_{0} z_{1} z_{3} z_{5} + (a_{11} t^{2} + a_{9} t) z_{0} z_{3}^{2} z_{5} + (a_{4} t) z_{1}^{2} z_{5}^{2} + (a_{10} t^{2} + a_{7} t) z_{1} z_{3} z_{5}^{2} + (a_{6} t^{2}) z_{3}^{2} z_{5}^{2}$
Top: ExtD4. Bottom: ExtD4.
F_0: I*_0. F_infinity: I*_0.
fib.set_random_coefficients()
fib.singular_fibers()

[ I_1 at -336.26, I_1 at -0.59812, I_1 at -0.56224 - 1.0260*I, I_1 at -0.56224 + 1.0260*I, I_1 at -0.36545 - 0.24903*I, I_1 at -0.36545 + 0.24903*I, I_1 at -0.15922 - 0.091813*I, I_1 at -0.15922 + 0.091813*I, I_1 at -0.096290, I_1 at -0.00090202, I_1 at 0.022784, I_1 at 2756.2, I*_0 at +Infinity, I*_0 at 0.00000 ]


Delta

3-d reflexive polytope #1529 in 3-d lattice M
print Delta.poly_x("")

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> TypeError: poly_x() takes at least 2 arguments (1 given) Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> TypeError: poly_x() takes at least 2 arguments (1 given)


d = dict()
for v in fib.coefficients_ring().gens():
d[v]=1
d[fib.coefficients_ring()("a12")] = fib.coefficients_ring()("a12")
d

{a4: 1, a5: 1, t: 1, a12: a12, a8: 1, a9: 1, a2: 1, a3: 1, a10: 1, a11: 1, a0: 1, a1: 1, a6: 1, a7: 1}
fib.set_coefficients(d)

fib.show()

A fibration of:
$\displaystyle \Delta^{3}_{2355}$
Corresponding to:
----------------------------------------------------------------------
2-dimensional reflexive slice of a 3-dimensional polytope. Points: [1, 2, 3, 6, 7, 13, 14] Skeleton points: [1, 2, 3, 6, 7, 13] Normal to the slice hyperplane: (1, -1, 0) Vertices of the slice polytope: M(-1, -1), M(-1, 0), M( 0, -1), M( 1, 1), M( 0, 1), M( 1, 0) in 2-d lattice M
----------------------------------------------------------------------
Coordinate points:
$\displaystyle \left(\begin{array}{rrrrrrrrrrrrrr} -1 & -1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 \\ 0 & -1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 & 0 & 1 & -1 & 1 & 0 & 0 & -1 & 1 & 0 & -1 \end{array}\right)$
Monomial points:
$\displaystyle \left(\begin{array}{rrrrrrrrrrrrr} -1 & -1 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & -1 & -1 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & -1 & 0 & 1 & -1 & 0 & 0 & -1 & 0 \end{array}\right)$
Equation of a hypersurface:
$\displaystyle z_{2}^{2} z_{3}^{2} z_{4}^{2} z_{5}^{2} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} + z_{0} z_{2}^{2} z_{3}^{2} z_{4} z_{5} z_{6} z_{7} z_{8}^{2} z_{11}^{2} z_{12} + z_{1} z_{2}^{2} z_{3} z_{4} z_{5}^{2} z_{7}^{2} z_{8} z_{9} z_{11}^{2} z_{12} + z_{0} z_{2} z_{3}^{2} z_{4}^{2} z_{5} z_{6}^{2} z_{8} z_{9} z_{10}^{2} z_{13} + z_{1} z_{2} z_{3} z_{4}^{2} z_{5}^{2} z_{6} z_{7} z_{9}^{2} z_{10}^{2} z_{13} + z_{0}^{2} z_{2} z_{3}^{2} z_{4} z_{6}^{2} z_{8}^{2} z_{10} z_{11} z_{12} z_{13} + a_{12} z_{0} z_{1} z_{2} z_{3} z_{4} z_{5} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} z_{12} z_{13} + z_{1}^{2} z_{2} z_{4} z_{5}^{2} z_{7}^{2} z_{9}^{2} z_{10} z_{11} z_{12} z_{13} + z_{0}^{2} z_{1} z_{2} z_{3} z_{6} z_{7} z_{8}^{2} z_{11}^{2} z_{12}^{2} z_{13} + z_{0} z_{1}^{2} z_{2} z_{5} z_{7}^{2} z_{8} z_{9} z_{11}^{2} z_{12}^{2} z_{13} + z_{0}^{2} z_{1} z_{3} z_{4} z_{6}^{2} z_{8} z_{9} z_{10}^{2} z_{12} z_{13}^{2} + z_{0} z_{1}^{2} z_{4} z_{5} z_{6} z_{7} z_{9}^{2} z_{10}^{2} z_{12} z_{13}^{2} + z_{0}^{2} z_{1}^{2} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} z_{12}^{2} z_{13}^{2}$