def a_number(self):
r"""
INPUT:
- ``E``: Hyperelliptic Curve of the form `y^2 = f(x)` over a finite field, `\mathbb{F}_q`
OUTPUT:
- ``a`` : a-number
EXAMPLES::
sage: K.<x>=GF(49,'x')[]
sage: C=HyperellipticCurve(x^5+1,0)
sage: C.a_number()
1
sage: K.<x>=GF(9,'x')[]
sage: C=HyperellipticCurve(x^7-1,0)
sage: C.a_number()
1
sage: P.<x>=GF(9,'a')[]
sage: E=HyperellipticCurve(x^29+1,0)
sage: E.a_number()
5
"""
#We use caching here since Cartier matrix is needed to compute a_number. So if the Cartier
#is already computed it is stored in list A. If it was not cached (i.e. A is empty), we simply
#compute it. If it is cached then we need to make sure that we have the correct one. So check
#which curve does the matrix correspond to. Since caching stores a lot of stuff, we only check
#the last entry in A. If it does not match, clear A and compute Cartier.
# Since Trac Ticket #11115, there is a special cache for methods
# that don't accept arguments. The easiest is: Call the cached
# method, and test whether the last entry is self.
M,Coeffs,g, Fq, p,E= self._Cartier_matrix_cached()
if E != self:
self._Cartier_matrix_cached.clear_cache()
M,Coeffs,g, Fq, p,E= self._Cartier_matrix_cached()
a=g-rank(M);
return a;
def a_number(self):
r"""
INPUT:
- ``E``: Hyperelliptic Curve of the form `y^2 = f(x)` over a finite field, `\mathbb{F}_q`
OUTPUT:
- ``a`` : a-number
EXAMPLES::
sage: K.<x>=GF(49,'x')[]
sage: C=HyperellipticCurve(x^5+1,0)
sage: C.a_number()
1
sage: K.<x>=GF(9,'x')[]
sage: C=HyperellipticCurve(x^7-1,0)
sage: C.a_number()
1
sage: P.<x>=GF(9,'a')[]
sage: E=HyperellipticCurve(x^29+1,0)
sage: E.a_number()
5
"""
#We use caching here since Cartier matrix is needed to compute a_number. So if the Cartier
#is already computed it is stored in list A. If it was not cached (i.e. A is empty), we simply
#compute it. If it is cached then we need to make sure that we have the correct one. So check
#which curve does the matrix correspond to. Since caching stores a lot of stuff, we only check
#the last entry in A. If it does not match, clear A and compute Cartier.
# Since Trac Ticket #11115, there is a special cache for methods
# that don't accept arguments. The easiest is: Call the cached
# method, and test whether the last entry is self.
M, Coeffs, g, Fq, p, E = self._Cartier_matrix_cached()
if E != self:
self._Cartier_matrix_cached.clear_cache()
M, Coeffs, g, Fq, p, E = self._Cartier_matrix_cached()
a = g - rank(M)
return a
def p_rank(self):
r"""
INPUT:
- ``E`` : Hyperelliptic Curve of the form `y^2 = f(x)` over a finite field, `\mathbb{F}_q`
OUTPUT:
- ``pr`` :p-rank
EXAMPLES::
sage: K.<x>=GF(49,'x')[]
sage: C=HyperellipticCurve(x^5+1,0)
sage: C.p_rank()
0
sage: K.<x>=GF(9,'x')[]
sage: C=HyperellipticCurve(x^7-1,0)
sage: C.p_rank()
0
sage: P.<x>=GF(9,'a')[]
sage: E=HyperellipticCurve(x^29+1,0)
sage: E.p_rank()
0
"""
#We use caching here since Hasse Witt is needed to compute p_rank. So if the Hasse Witt
#is already computed it is stored in list A. If it was not cached (i.e. A is empty), we simply
#compute it. If it is cached then we need to make sure that we have the correct one. So check
#which curve does the matrix correspond to. Since caching stores a lot of stuff, we only check
#the last entry in A. If it does not match, clear A and compute Hasse Witt.
# However, it seems a waste of time to manually analyse the cache
# -- See Trac Ticket #11115
N,E= self._Hasse_Witt_cached()
if E!=self:
self._Hasse_Witt_cached.clear_cache()
N,E= self._Hasse_Witt_cached()
pr=rank(N);
return pr
def p_rank(self):
r"""
INPUT:
- ``E`` : Hyperelliptic Curve of the form `y^2 = f(x)` over a finite field, `\mathbb{F}_q`
OUTPUT:
- ``pr`` :p-rank
EXAMPLES::
sage: K.<x>=GF(49,'x')[]
sage: C=HyperellipticCurve(x^5+1,0)
sage: C.p_rank()
0
sage: K.<x>=GF(9,'x')[]
sage: C=HyperellipticCurve(x^7-1,0)
sage: C.p_rank()
0
sage: P.<x>=GF(9,'a')[]
sage: E=HyperellipticCurve(x^29+1,0)
sage: E.p_rank()
0
"""
#We use caching here since Hasse Witt is needed to compute p_rank. So if the Hasse Witt
#is already computed it is stored in list A. If it was not cached (i.e. A is empty), we simply
#compute it. If it is cached then we need to make sure that we have the correct one. So check
#which curve does the matrix correspond to. Since caching stores a lot of stuff, we only check
#the last entry in A. If it does not match, clear A and compute Hasse Witt.
# However, it seems a waste of time to manually analyse the cache
# -- See Trac Ticket #11115
N, E = self._Hasse_Witt_cached()
if E != self:
self._Hasse_Witt_cached.clear_cache()
N, E = self._Hasse_Witt_cached()
pr = rank(N)
return pr
def _set_rank(self, cd):
r"""
Returns the rank of degree `cd` as well as the matrix of relations.
For given codimension cd, returns dimension of kernel of relations which is rank of MW^cd and matrix A of relations
Any vector in the kernel of A is a balanced weight of codimension cd
TESTS::
sage: vertices = [(0,1,0),(0,1,1),(0,2,0),(0,2,2),(1,1,1),(1,2,1),(1,2,2)]
sage: fan = NormalFan(Polyhedron(vertices = vertices))
sage: A = MW(fan)
sage: A._set_rank(2)
(
[ 0 0 1 -1 0 -1]
[ 1 -1 0 0 1 0]
3, [-1 0 0 1 0 0]
)
sage: A._set_rank(5)
(0, [])
"""
d = self._dim - cd
ConeList = self._cones(d) #it's a tuple!
n = len(ConeList)
#generate balancing conditions for every cone of dimension d-1
ConeRelns = self._cones(d - 1)
relations = []
for c in ConeRelns:
balance = self._balancing(c)
for b in balance:
relations.append(b)
# remove redundant relations, rank = dimension of space perpendicular to the list of vectors
A = Matrix(relations)
r = rank(A)
# kernel(A.T) gives a basis for MW^cd where coordinates designate coefficients of cones in that place
return n - r, A