78 KiB
78 KiB
class superelliptic:
def __init__(self, f, m):
Rx = f.parent()
x = Rx.gen()
F = Rx.base()
Rx.<x> = PolynomialRing(F)
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
self.polynomial = Rx(f)
self.exponent = m
self.base_ring = F
self.characteristic = F.characteristic()
r = Rx(f).degree()
delta = GCD(r, m)
def __repr__(self):
f = self.polynomial
m = self.exponent
F = self.base_ring
return 'Superelliptic curve with the equation y^' + str(m) + ' = ' + str(f)+' over ' + str(F)
def basis_holomorphic_differentials_degree(self):
f = self.polynomial
m = self.exponent
r = f.degree()
delta = GCD(r, m)
F = self.base_ring
Rx.<x> = PolynomialRing(F)
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
#########basis of holomorphic differentials and de Rham
basis_holo = []
degrees0 = {}
k = 0
for j in range(1, m):
for i in range(1, r):
if (r*j - m*i >= delta):
basis_holo += [superelliptic_form(self, Fxy(x^(i-1)/y^j))]
degrees0[k] = (i-1, j)
k = k+1
return(basis_holo, degrees0)
def holomorphic_differentials_basis(self):
basis_holo, degrees0 = self.basis_holomorphic_differentials_degree()
return basis_holo
def degrees_holomorphic_differentials(self):
basis_holo, degrees0 = self.basis_holomorphic_differentials_degree()
return degrees0
def basis_de_rham_degrees(self):
f = self.polynomial
m = self.exponent
r = f.degree()
delta = GCD(r, m)
F = self.base_ring
Rx.<x> = PolynomialRing(F)
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
basis_holo = self.holomorphic_differentials_basis()
basis = []
for k in range(0, len(basis_holo)):
basis += [superelliptic_cech(self, basis_holo[k], superelliptic_function(self, 0))]
## non-holomorphic elts of H^1_dR
t = len(basis)
degrees0 = {}
degrees1 = {}
for j in range(1, m):
for i in range(1, r):
if (r*(m-j) - m*i >= delta):
s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f
psi = Rx(cut(s, i))
basis += [superelliptic_cech(self, superelliptic_form(self, Fxy(psi/y^j)), superelliptic_function(self, Fxy(m*y^(m-j)/x^i)))]
degrees0[t] = (psi.degree(), j)
degrees1[t] = (-i, m-j)
t += 1
return basis, degrees0, degrees1
def de_rham_basis(self):
basis, degrees0, degrees1 = self.basis_de_rham_degrees()
return basis
def degrees_de_rham0(self):
basis, degrees0, degrees1 = self.basis_de_rham_degrees()
return degrees0
def degrees_de_rham1(self):
basis, degrees0, degrees1 = self.basis_de_rham_degrees()
return degrees1
def is_smooth(self):
f = self.polynomial
if f.discriminant() == 0:
return 0
return 1
def genus(self):
r = self.polynomial.degree()
m = self.exponent
delta = GCD(r, m)
return 1/2*((r-1)*(m-1) - delta + 1)
def verschiebung_matrix(self):
basis = self.de_rham_basis()
g = self.genus()
p = self.characteristic
F = self.base_ring
M = matrix(F, 2*g, 2*g)
for i in range(0, len(basis)):
w = basis[i]
v = w.verschiebung().coordinates()
M[i, :] = v
return M
def frobenius_matrix(self):
basis = self.de_rham_basis()
g = self.genus()
p = self.characteristic
F = self.base_ring
M = matrix(F, 2*g, 2*g)
for i in range(0, len(basis)):
w = basis[i]
v = w.frobenius().coordinates()
M[i, :] = v
return M
def cartier_matrix(self):
basis = self.holomorphic_differentials_basis()
g = self.genus()
p = self.characteristic
F = self.base_ring
M = matrix(F, g, g)
for i in range(0, len(basis)):
w = basis[i]
v = w.cartier().coordinates()
M[i, :] = v
return M
# def p_rank(self):
# return self.cartier_matrix().rank()
def a_number(self):
g = C.genus()
return g - self.cartier_matrix().rank()
def final_type(self, test = 0):
Fr = self.frobenius_matrix()
V = self.verschiebung_matrix()
p = self.characteristic
return flag(Fr, V, p, test)
def reduction(C, g):
p = C.characteristic
F = C.base_ring
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
f = C.polynomial
r = f.degree()
m = C.exponent
g = Fxy(g)
g1 = g.numerator()
g2 = g.denominator()
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
(A, B, C) = xgcd(FxRy(g2), FxRy(y^m - f))
g = FxRy(g1*B/A)
while(g.degree(Rxy(y)) >= m):
d = g.degree(Rxy(y))
G = coff(g, d)
i = floor(d/m)
g = g - G*y^d + f^i * y^(d%m) *G
return(FxRy(g))
def reduction_form(C, g):
F = C.base_ring
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
f = C.polynomial
r = f.degree()
m = C.exponent
g = reduction(C, g)
g1 = Rxy(0)
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = FxRy(g)
for j in range(0, m):
if j==0:
G = coff(g, 0)
g1 += FxRy(G)
else:
G = coff(g, j)
g1 += Fxy(y^(j-m)*f*G)
return(g1)
class superelliptic_function:
def __init__(self, C, g):
F = C.base_ring
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
f = C.polynomial
r = f.degree()
m = C.exponent
self.curve = C
g = reduction(C, g)
self.function = g
def __repr__(self):
return str(self.function)
def jth_component(self, j):
g = self.function
C = self.curve
F = C.base_ring
Rx.<x> = PolynomialRing(F)
Fx.<x> = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = FxRy(g)
return coff(g, j)
def __add__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 + g2)
return superelliptic_function(C, g)
def __sub__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 - g2)
return superelliptic_function(C, g)
def __mul__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 * g2)
return superelliptic_function(C, g)
def __truediv__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 / g2)
return superelliptic_function(C, g)
def diffn(self):
C = self.curve
f = C.polynomial
m = C.exponent
F = C.base_ring
g = self.function
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
g = Fxy(g)
A = g.derivative(x)
B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))
return superelliptic_form(C, A+B)
def expansion_at_infty(self, i = 0, prec=10):
C = self.curve
f = C.polynomial
m = C.exponent
F = C.base_ring
Rx.<x> = PolynomialRing(F)
f = Rx(f)
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RptW.<W> = PolynomialRing(Rt)
RptWQ = FractionField(RptW)
Rxy.<x, y> = PolynomialRing(F)
RxyQ = FractionField(Rxy)
fct = self.function
fct = RxyQ(fct)
r = f.degree()
delta, a, b = xgcd(m, r)
b = -b
M = m/delta
R = r/delta
while a<0:
a += R
b += M
g = (x^r*f(x = 1/x))
gW = RptWQ(g(x = t^M * W^b)) - W^(delta)
ww = naive_hensel(gW, F, start = root_of_unity(F, delta)^i, prec = prec)
xx = Rt(1/(t^M*ww^b))
yy = 1/(t^R*ww^a)
return Rt(fct(x = Rt(xx), y = Rt(yy)))
def naive_hensel(fct, F, start = 1, prec=10):
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
RptW.<W> = PolynomialRing(RtQ)
RptWQ = FractionField(RptW)
fct = RptWQ(fct)
fct = RptW(numerator(fct))
#return(fct)
#while fct not in RptW:
# print(fct)
# fct *= W
alpha = (fct.derivative())(W = start)
w0 = Rt(start)
i = 1
while(i < prec):
w0 = w0 - fct(W = w0)/alpha + O(t^(prec))
i += 1
return w0
class superelliptic_form:
def __init__(self, C, g):
F = C.base_ring
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
g = Fxy(reduction_form(C, g))
self.form = g
self.curve = C
def __add__(self, other):
C = self.curve
g1 = self.form
g2 = other.form
g = reduction(C, g1 + g2)
return superelliptic_form(C, g)
def __sub__(self, other):
C = self.curve
g1 = self.form
g2 = other.form
g = reduction(C, g1 - g2)
return superelliptic_form(C, g)
def __repr__(self):
g = self.form
if len(str(g)) == 1:
return str(g) + ' dx'
return '('+str(g) + ') dx'
def __rmul__(self, constant):
C = self.curve
omega = self.form
return superelliptic_form(C, constant*omega)
def cartier(self):
C = self.curve
m = C.exponent
p = C.characteristic
f = C.polynomial
F = C.base_ring
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
Fxy = FractionField(FxRy)
result = superelliptic_form(C, FxRy(0))
mult_order = Integers(m)(p).multiplicative_order()
M = Integer((p^(mult_order)-1)/m)
for j in range(1, m):
fct_j = self.jth_component(j)
h = Rx(fct_j*f^(M*j))
j1 = (p^(mult_order-1)*j)%m
B = floor(p^(mult_order-1)*j/m)
result += superelliptic_form(C, polynomial_part(p, h)/(f^B*y^(j1)))
return result
def coordinates(self):
C = self.curve
F = C.base_ring
m = C.exponent
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = C.genus()
degrees_holo = C.degrees_holomorphic_differentials()
degrees_holo_inv = {b:a for a, b in degrees_holo.items()}
basis = C.holomorphic_differentials_basis()
for j in range(1, m):
omega_j = Fx(self.jth_component(j))
if omega_j != Fx(0):
d = degree_of_rational_fctn(omega_j, F)
index = degrees_holo_inv[(d, j)]
a = coeff_of_rational_fctn(omega_j, F)
a1 = coeff_of_rational_fctn(basis[index].jth_component(j), F)
elt = self - (a/a1)*basis[index]
return elt.coordinates() + a/a1*vector([F(i == index) for i in range(0, g)])
return vector(g*[0])
def jth_component(self, j):
g = self.form
C = self.curve
F = C.base_ring
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
Fxy = FractionField(FxRy)
Ryinv.<y_inv> = PolynomialRing(Fx)
g = Fxy(g)
g = g(y = 1/y_inv)
g = Ryinv(g)
return coff(g, j)
def is_regular_on_U0(self):
C = self.curve
F = C.base_ring
m = C.exponent
Rx.<x> = PolynomialRing(F)
for j in range(1, m):
if self.jth_component(j) not in Rx:
return 0
return 1
def is_regular_on_Uinfty(self):
C = self.curve
F = C.base_ring
m = C.exponent
f = C.polynomial
r = f.degree()
delta = GCD(m, r)
M = m/delta
R = r/delta
for j in range(1, m):
A = self.jth_component(j)
d = degree_of_rational_fctn(A, F)
if(-d*M + j*R -(M+1)<0):
return 0
return 1
def expansion_at_infty(self, i = 0, prec=10):
g = self.form
C = self.curve
g = superelliptic_function(C, g)
g = g.expansion_at_infty(i = i, prec=prec)
x_series = superelliptic_function(C, x).expansion_at_infty(i = i, prec=prec)
dx_series = x_series.derivative()
return g*dx_series
class superelliptic_cech:
def __init__(self, C, omega, fct):
self.omega0 = omega
self.omega8 = omega - fct.diffn()
self.f = fct
self.curve = C
def __add__(self, other):
C = self.curve
return superelliptic_cech(C, self.omega0 + other.omega0, self.f + other.f)
def __sub__(self, other):
C = self.curve
return superelliptic_cech(C, self.omega0 - other.omega0, self.f - other.f)
def __rmul__(self, constant):
C = self.curve
w1 = self.omega0.form
f1 = self.f.function
w2 = superelliptic_form(C, constant*w1)
f2 = superelliptic_function(C, constant*f1)
return superelliptic_cech(C, w2, f2)
def __repr__(self):
return "(" + str(self.omega0) + ", " + str(self.f) + ", " + str(self.omega8) + ")"
def verschiebung(self):
C = self.curve
omega = self.omega0
F = C.base_ring
Rx.<x> = PolynomialRing(F)
return superelliptic_cech(C, omega.cartier(), superelliptic_function(C, Rx(0)))
def frobenius(self):
C = self.curve
fct = self.f.function
p = C.characteristic
Rx.<x> = PolynomialRing(F)
return superelliptic_cech(C, superelliptic_form(C, Rx(0)), superelliptic_function(C, fct^p))
def coordinates(self):
C = self.curve
F = C.base_ring
m = C.exponent
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = C.genus()
degrees_holo = C.degrees_holomorphic_differentials()
degrees_holo_inv = {b:a for a, b in degrees_holo.items()}
degrees0 = C.degrees_de_rham0()
degrees0_inv = {b:a for a, b in degrees0.items()}
degrees1 = C.degrees_de_rham1()
degrees1_inv = {b:a for a, b in degrees1.items()}
basis = C.de_rham_basis()
omega = self.omega0
fct = self.f
if fct.function == Rx(0) and omega.form != Rx(0):
for j in range(1, m):
omega_j = Fx(omega.jth_component(j))
if omega_j != Fx(0):
d = degree_of_rational_fctn(omega_j, F)
index = degrees_holo_inv[(d, j)]
a = coeff_of_rational_fctn(omega_j, F)
a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j), F)
elt = self - (a/a1)*basis[index]
return elt.coordinates() + a/a1*vector([F(i == index) for i in range(0, 2*g)])
for j in range(1, m):
fct_j = Fx(fct.jth_component(j))
if (fct_j != Rx(0)):
d = degree_of_rational_fctn(fct_j, p)
if (d, j) in degrees1.values():
index = degrees1_inv[(d, j)]
a = coeff_of_rational_fctn(fct_j, F)
elt = self - (a/m)*basis[index]
return elt.coordinates() + a/m*vector([F(i == index) for i in range(0, 2*g)])
if d<0:
a = coeff_of_rational_fctn(fct_j, F)
h = superelliptic_function(C, FxRy(a*y^j*x^d))
elt = superelliptic_cech(C, self.omega0, self.f - h)
return elt.coordinates()
if (fct_j != Rx(0)):
G = superelliptic_function(C, y^j*x^d)
a = coeff_of_rational_fctn(fct_j, F)
elt =self - a*superelliptic_cech(C, diffn(G), G)
return elt.coordinates()
return vector(2*g*[0])
def is_cocycle(self):
w0 = self.omega0
w8 = self.omega8
fct = self.f
if not w0.is_regular_on_U0() and not w8.is_regular_on_Uinfty():
return('w0 & w8')
if not w0.is_regular_on_U0():
return('w0')
if not w8.is_regular_on_Uinfty():
return('w8')
if w0.is_regular_on_U0() and w8.is_regular_on_Uinfty():
return 1
return 0
def degree_of_rational_fctn(f, F):
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
f = Fx(f)
f1 = f.numerator()
f2 = f.denominator()
d1 = f1.degree()
d2 = f2.degree()
return(d1 - d2)
def coeff_of_rational_fctn(f, F):
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
f = Fx(f)
if f == Rx(0):
return 0
f1 = f.numerator()
f2 = f.denominator()
d1 = f1.degree()
d2 = f2.degree()
a1 = f1.coefficients(sparse = false)[d1]
a2 = f2.coefficients(sparse = false)[d2]
return(a1/a2)
def coff(f, d):
lista = f.coefficients(sparse = false)
if len(lista) <= d:
return 0
return lista[d]
def cut(f, i):
R = f.parent()
coeff = f.coefficients(sparse = false)
return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))
def polynomial_part(p, h):
F = GF(p)
Rx.<x> = PolynomialRing(F)
h = Rx(h)
result = Rx(0)
for i in range(0, h.degree()+1):
if (i%p) == p-1:
power = Integer((i-(p-1))/p)
result += Integer(h[i]) * x^(power)
return result
#Find delta-th root of unity in field F
def root_of_unity(F, delta):
Rx.<x> = PolynomialRing(F)
cyclotomic = x^(delta) - 1
for root, a in cyclotomic.roots():
powers = [root^d for d in delta.divisors() if d!= delta]
if 1 not in powers:
return root
def preimage(U, V, M): #preimage of subspace U under M
basis_preimage = M.right_kernel().basis()
imageU = U.intersection(M.transpose().image())
basis = imageU.basis()
for v in basis:
w = M.solve_right(v)
basis_preimage = basis_preimage + [w]
return V.subspace(basis_preimage)
def image(U, V, M):
basis = U.basis()
basis_image = []
for v in basis:
basis_image += [M*v]
return V.subspace(basis_image)
def flag(F, V, p, test = 0):
dim = F.dimensions()[0]
space = VectorSpace(GF(p), dim)
flag_subspaces = (dim+1)*[0]
flag_used = (dim+1)*[0]
final_type = (dim+1)*['?']
flag_subspaces[dim] = space
flag_used[dim] = 1
while 1 in flag_used:
index = flag_used.index(1)
flag_used[index] = 0
U = flag_subspaces[index]
U_im = image(U, space, V)
d_im = U_im.dimension()
final_type[index] = d_im
U_pre = preimage(U, space, F)
d_pre = U_pre.dimension()
if flag_subspaces[d_im] == 0:
flag_subspaces[d_im] = U_im
flag_used[d_im] = 1
if flag_subspaces[d_pre] == 0:
flag_subspaces[d_pre] = U_pre
flag_used[d_pre] = 1
if test == 1:
print('(', final_type, ')')
for i in range(0, dim+1):
if final_type[i] == '?' and final_type[dim - i] != '?':
i1 = dim - i
final_type[i] = final_type[i1] - i1 + dim/2
final_type[0] = 0
for i in range(1, dim+1):
if final_type[i] == '?':
prev = final_type[i-1]
if prev != '?' and prev in final_type[i+1:]:
final_type[i] = prev
for i in range(1, dim+1):
if final_type[i] == '?':
final_type[i] = min(final_type[i-1] + 1, dim/2)
if is_final(final_type, dim/2):
return final_type[1:dim/2 + 1]
print('error:', final_type[1:dim/2 + 1])
def is_final(final_type, dim):
n = len(final_type)
if final_type[0] != 0:
return 0
if final_type[n-1] != dim:
return 0
for i in range(1, n):
if final_type[i] != final_type[i - 1] and final_type[i] != final_type[i - 1] + 1:
return 0
return 1class as_cover:
def __init__(self, C, list_of_fcts, prec = 10):
self.quotient = C
self.functions = list_of_fcts
self.height = len(list_of_fcts)
p = C.characteristic
self.characteristic = p
self.prec = prec
f = C.polynomial
m = C.exponent
r = f.degree()
delta = GCD(m, r)
self.nb_of_pts_at_infty = delta
Rxy.<x, y> = PolynomialRing(GF(p), 2)
Rt.<t> = LaurentSeriesRing(GF(p), default_prec=prec)
all_x_series = []
all_y_series = []
all_z_series = []
all_dx_series = []
all_jumps = []
for i in range(delta):
x_series = superelliptic_function(C, x).expansion_at_infty(i = i, prec=prec)
y_series = superelliptic_function(C, y).expansion_at_infty(i = i, prec=prec)
z_series = []
jumps = []
n = len(list_of_fcts)
list_of_power_series = [g.expansion_at_infty(i = i, prec=prec) for g in list_of_fcts]
for i in range(n):
power_series = list_of_power_series[i]
jump, correction, t_old, z = artin_schreier_transform(power_series, prec = prec)
x_series = x_series(t = t_old)
y_series = y_series(t = t_old)
z_series = [zi(t = t_old) for zi in z_series]
z_series += [z]
jumps += [jump]
list_of_power_series = [g(t = t_old) for g in list_of_power_series]
all_jumps += [jumps]
all_x_series += [x_series]
all_y_series += [y_series]
all_z_series += [z_series]
all_dx_series += [x_series.derivative()]
self.jumps = all_jumps
self.x = all_x_series
self.y = all_y_series
self.z = all_z_series
self.dx = all_dx_series
def __repr__(self):
n = self.height
p = self.characteristic
if n==1:
return "(Z/p)-cover of " + str(self.quotient)+" with the equation:\n z^" + str(p) + " - z = " + str(self.functions[0])
result = "(Z/p)^"+str(self.height)+ "-cover of " + str(self.quotient)+" with the equations:\n"
for i in range(n):
result += 'z' + str(i) + "^" + str(p) + " - z" + str(i) + " = " + str(self.functions[i]) + "\n"
return result
def genus(self):
jumps = self.jumps
gY = self.quotient.genus()
n = self.height
delta = self.nb_of_pts_at_infty
return p*gY + (p^n - 1)*(delta - 1) + sum(p^(n-j-1)*(jumps[i][j]-1)*(p-1)/2 for j in range(n) for i in range(delta))
def exponent_of_different(self, i = 0):
jumps = self.jumps
n = self.height
delta = self.nb_of_pts_at_infty
return sum(p^(n-j-1)*(jumps[i][j]+1)*(p-1) for j in range(n))
def holomorphic_differentials_basis(self):
from itertools import product
x_series = self.x
y_series = self.y
z_series = self.z
dx_series = self.dx
delta = self.nb_of_pts_at_infty
p = self.characteristic
n = self.height
prec = self.prec
C = self.quotient
m = C.exponent
r = C.polynomial.degree()
variable_names = 'x, y'
for i in range(n):
variable_names += ', z' + str(i)
Rxyz = PolynomialRing(GF(p), n+2, variable_names)
x, y = Rxyz.gens()[:2]
z = Rxyz.gens()[2:]
RxyzQ = FractionField(Rxyz)
Rt.<t> = LaurentSeriesRing(GF(p), default_prec=prec)
#Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
S = []
RQxyz = FractionField(Rxyz)
pr = [list(GF(p)) for _ in range(n)]
for i in range(0, 3*r):
for j in range(0, m):
for k in product(*pr):
eta = as_form(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)
eta_exp = eta.expansion_at_infty()
S += [(eta, eta_exp)]
forms = holomorphic_combinations(S)
for i in range(1, delta):
forms = [(omega, omega.expansion_at_infty(i = i)) for omega in forms]
forms = holomorphic_combinations(forms)
if len(forms) < self.genus():
print("I haven't found all forms.")
return forms
#given a set S of (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt
def holomorphic_combinations(S):
C_AS = S[0][0].curve
p = C_AS.characteristic
F = GF(p)
prec = C_AS.prec
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
minimal_valuation = min([g[1].valuation() for g in S])
if minimal_valuation >= 0:
return [s[0] for s in S]
list_of_lists = [] #to będzie lista złożona z list współczynników część nieholomorficznych rozwinięcia form z S
for eta, eta_exp in S:
a = -minimal_valuation + eta_exp.valuation()
list_coeffs = a*[0] + eta_exp.list() + (-minimal_valuation)*[0]
list_coeffs = list_coeffs[:-minimal_valuation]
list_of_lists += [list_coeffs]
M = matrix(GF(p), list_of_lists)
V = M.kernel() #chcemy wyzerować części nieholomorficzne, biorąc kombinacje form z S
# Sprawdzamy, jakim formom odpowiadają elementy V.
forms = []
for vec in V.basis():
forma_holo = as_form(C_AS, 0)
forma_holo_power_series = Rt(0)
for vec_wspolrzedna, elt_S in zip(vec, S):
eta = elt_S[0]
#eta_exp = elt_S[1]
forma_holo += vec_wspolrzedna*eta
#forma_holo_power_series += vec_wspolrzedna*eta_exp
forms += [forma_holo]
return forms
class as_function:
def __init__(self, C, g):
self.curve = C
n = C.height
variable_names = 'x, y'
for i in range(n):
variable_names += ', z' + str(i)
Rxyz = PolynomialRing(GF(p), n+2, variable_names)
x, y = Rxyz.gens()[:2]
z = Rxyz.gens()[2:]
RxyzQ = FractionField(Rxyz)
self.function = RxyzQ(g)
class as_form:
def __init__(self, C, g):
self.curve = C
n = C.height
variable_names = 'x, y'
for i in range(n):
variable_names += ', z' + str(i)
Rxyz = PolynomialRing(GF(p), n+2, variable_names)
x, y = Rxyz.gens()[:2]
z = Rxyz.gens()[2:]
RxyzQ = FractionField(Rxyz)
self.form = RxyzQ(g)
def __repr__(self):
return "(" + str(self.form)+") * dx"
def expansion_at_infty(self, i = 0):
C = self.curve
delta = C.nb_of_pts_at_infty
x_series = C.x[i]
y_series = C.y[i]
z_series = C.z[i]
dx_series = C.dx[i]
n = C.height
variable_names = 'x, y'
for j in range(n):
variable_names += ', z' + str(j)
Rxyz = PolynomialRing(GF(p), n+2, variable_names)
x, y = Rxyz.gens()[:2]
z = Rxyz.gens()[2:]
RxyzQ = FractionField(Rxyz)
prec = C.prec
Rt.<t> = LaurentSeriesRing(GF(p), default_prec=prec)
g = self.form
sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
return g.substitute(sub_list)*dx_series
def __add__(self, other):
C = self.curve
g1 = self.form
g2 = other.form
return as_form(C, g1 + g2)
def __sub__(self, other):
C = self.curve
g1 = self.form
g2 = other.form
return as_form(C, g1 - g2)
def __rmul__(self, constant):
C = self.curve
omega = self.form
return as_form(C, constant*omega)
def group_action(self, ZN_tuple):
C = self.curve
n = C.height
variable_names = 'x, y'
for j in range(n):
variable_names += ', z' + str(j)
Rxyz = PolynomialRing(GF(p), n+2, variable_names)
x, y = Rxyz.gens()[:2]
z = Rxyz.gens()[2:]
RxyzQ = FractionField(Rxyz)
sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}
g = self.form
return as_form(C, g.substitute(sub_list))
def coordinates(self):
C = self.curve
n = C.height
gC = C.genus()
holo = C.holomorphic_differentials_basis()
nb_of_coeffs = 2*gC
holo_exp = [omega.expansion_at_infty() for omega in holo]
list_of_coeffs = []
for omega in holo_exp:
a = omega.valuation()
added_list = list(omega)[:nb_of_coeffs]
added_list = a*[0] + added_list
added_list = added_list[:nb_of_coeffs]
list_of_coeffs += [added_list]
M = matrix(GF(p), list_of_coeffs)
M = M.transpose()
omega = self.expansion_at_infty()
a = omega.valuation()
v = list(omega)[:nb_of_coeffs]
v = a*[0] + v
v = v[:nb_of_coeffs]
v = vector(v)
v1 = M.solve_right(v)
return v1
def val_magical(C_AS):
n = C_AS.height
variable_names = 'x, y'
for i in range(n):
variable_names += ', z' + str(i)
Rxyz = PolynomialRing(GF(p), n+2, variable_names)
x, y = Rxyz.gens()[:2]
z = Rxyz.gens()[2:]
RxyzQ = FractionField(Rxyz)
C = C_AS.quotient
forms = C.holomorphic_differentials_basis()
z_mag = prod(z[i]^(p-1) for i in range(n))
forms_list = [z_mag*RxyzQ(omega.form) for omega in forms]
forms_list = [as_form(C_AS, omega) for omega in forms_list]
forms_list = [omega.expansion_at_infty() for omega in forms_list]
forms_list = [omega.valuation() for omega in forms_list]
return min(forms_list)
# Given power_series, find its reverse (g with g \circ power_series = id) with given precision
def new_reverse(power_series, prec = 10):
F = power_series.parent().base()
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
power_series = RtQ(power_series)
a = power_series.list()[0]
g = 1/a*t
n = 2
while(n <= prec):
aux = power_series(t = g) - t
if aux.valuation() > n:
b = 0
else:
b = aux.list()[0]
g = g - b/a*t^n
n += 1
return g
# Given a power_series, find correction such that power_series - (correction)^p +correction has valuation
# -jump non divisible by p. Also, express t (the variable) in terms of the uniformizer at infty on the curve
# z^p - z = power_series, where z = 1/t_new^(jump) and express z in terms of the new uniformizer.
def artin_schreier_transform(power_series, prec = 10):
correction = 0
F = power_series.parent().base()
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
power_series = RtQ(power_series)
if power_series.valuation() == +Infinity:
return(0,0,t,0)
while(power_series.valuation() % p == 0 and power_series.valuation() < 0):
M = -power_series.valuation()/p
coeff = power_series.list()[0] #wspolczynnik a_(-p) w f_AS
correction += coeff*t^(-M)
power_series = power_series - coeff*(t^(-p*M) - t^(-M))
jump = max(-(power_series.valuation()), 0)
try:
T = ((power_series)^(-1)).nth_root(jump) #T is defined by power_series = 1/T^m
except:
print("no ", str(jump), "-th root; divide by", power_series.list()[0])
return (jump, power_series.list()[0])
T_rev = new_reverse(T, prec = prec)
t_old = T_rev(t^p/(1 - t^((p-1)*jump)).nth_root(jump))
z = 1/t^(jump) + Rt(correction)(t = t_old)
return(jump, correction, t_old, z)
def are_forms_linearly_dependent(set_of_forms):
from sage.rings.polynomial.toy_variety import is_linearly_dependent
C = set_of_forms[0].curve
n = C.height
variable_names = 'x, y'
for i in range(n):
variable_names += ', z' + str(i)
Rxyz = PolynomialRing(GF(p), n+2, variable_names)
denominators = prod(denominator(omega.form) for omega in set_of_forms)
return is_linearly_dependent([Rxyz(denominators*omega.form) for omega in set_of_forms])
def group_action_matrices(C_AS):
holo = C_AS.holomorphic_differentials_basis()
n = C_AS.height
A = [[] for i in range(n)]
for omega in holo:
for i in range(n):
ei = n*[0]
ei[i] = 1
omega1 = omega.group_action(ei)
v1 = omega1.coordinates()
A[i] += [v1]
for i in range(n):
A[i] = matrix(GF(p), A[i])
A[i] = A[i].transpose()
return ATesty
p = 5
m = 2
Rx.<x> = PolynomialRing(GF(p))
f = x^3 + x^2 + 1
C_super = superelliptic(f, m)
Rxy.<x, y> = PolynomialRing(GF(p), 2)
fArS1 = superelliptic_function(C_super, y*x)
fArS2 = superelliptic_function(C_super, y*x^2)
fArS3 = superelliptic_function(C_super, y)
AS1 = as_cover(C_super, [fArS1, fArS2, fArS3], prec=500)
print(AS1.genus())
AS2 = as_cover(C_super, [fArS2, fArS3, fArS1], prec=500)
print(AS2.genus())325 325
p = 5
m = 2
Rx.<x> = PolynomialRing(GF(p))
f = x^3 + x^2 + 1
C_super = superelliptic(f, m)
Rxy.<x, y> = PolynomialRing(GF(p), 2)
fArS1 = superelliptic_function(C_super, y*x)
fArS2 = superelliptic_function(C_super, y*x^2)
fArS3 = superelliptic_function(C_super, y)
AS1 = as_cover(C_super, [fArS1, fArS2, fArS3], prec=1000)
print(AS1.exponent_of_different())
omega = as_form(AS1, 1/y)
print(omega.expansion_at_infty())Brudnopis
p = 11
m = 2
Rx.<x> = PolynomialRing(GF(p))
f = x^7 + x^2 + 1
C_super = superelliptic(f, m)
print(C_super.genus())3
Rxy.<x, y> = PolynomialRing(GF(p))
fArS1 = superelliptic_function(C_super, y*x^2)
AS = as_cover(C_super, [fArS1], prec=300)
print(AS.genus())
omega = C_super.holomorphic_differentials_basis()[2]
magical_for_a_form(omega, AS)33
[(x^2/y) * dx, (x^2*z0/y) * dx, ((x^2*z0^2 - x^3)/y) * dx]
AS.genus()33
p = 11
m = 2
Rx.<x> = PolynomialRing(GF(p))
f = x^7 + x^2 + 1
C_super = superelliptic(f, m)
print(C_super.genus())
Rxy.<x, y> = PolynomialRing(GF(p))
for a in GF(p):
for b in GF(p):
for c in GF(p):
if (b != GF(p)(0) or a!=GF(p)(0)):
f1 = superelliptic_function(C_super, y*x^2)
f2 = superelliptic_function(C_super, a*y+b*x+c*x^2)
try:
AS = as_cover(C_super, [f1, f2], prec=500)
if AS.jumps[1] >0 and AS.jumps[1] < 67:
print(a, b, c, AS.jumps, AS.genus())
except:
pass3 0 0 2 [1, 34] 198 0 0 6 [1, 34] 198 0 0 7 [1, 34] 198 0 0 8 [1, 34] 198 0 0 10 [1, 34] 198 0 1 0 [1, 12] 88 0 1 6 [1, 34] 198 0 1 7 [1, 34] 198 0 1 8 [1, 34] 198 0 1 10 [1, 34] 198 0 2 2 [1, 34] 198 0 2 6 [1, 34] 198 0 2 7 [1, 34] 198 0 3 2 [1, 34] 198
[0;31m---------------------------------------------------------------------------[0m [0;31mKeyboardInterrupt[0m Traceback (most recent call last) Input [0;32mIn [22][0m, in [0;36m<cell line: 9>[0;34m()[0m [1;32m 10[0m [38;5;28;01mfor[39;00m b [38;5;129;01min[39;00m GF(p): [1;32m 11[0m [38;5;28;01mfor[39;00m c [38;5;129;01min[39;00m GF(p): [0;32m---> 12[0m f1 [38;5;241m=[39m [43msuperelliptic_function[49m[43m([49m[43mC_super[49m[43m,[49m[43m [49m[43my[49m[38;5;241;43m*[39;49m[43mx[49m[38;5;241;43m*[39;49m[38;5;241;43m*[39;49m[43mInteger[49m[43m([49m[38;5;241;43m2[39;49m[43m)[49m[43m)[49m [1;32m 13[0m f2 [38;5;241m=[39m superelliptic_function(C_super, a[38;5;241m*[39my[38;5;241m+[39mb[38;5;241m*[39mx[38;5;241m+[39mc[38;5;241m*[39mx[38;5;241m*[39m[38;5;241m*[39mInteger([38;5;241m2[39m)) [1;32m 14[0m [38;5;28;01mtry[39;00m: Input [0;32mIn [2][0m, in [0;36msuperelliptic_function.__init__[0;34m(self, C, g)[0m [1;32m 209[0m [38;5;28;01mclass[39;00m [38;5;21;01msuperelliptic_function[39;00m: [0;32m--> 210[0m [38;5;28;01mdef[39;00m [38;5;21m__init__[39m([38;5;28mself[39m, C, g): [1;32m 211[0m F [38;5;241m=[39m C[38;5;241m.[39mbase_ring [1;32m 212[0m Rxy [38;5;241m=[39m PolynomialRing(F, Integer([38;5;241m2[39m), names[38;5;241m=[39m([38;5;124m'[39m[38;5;124mx[39m[38;5;124m'[39m, [38;5;124m'[39m[38;5;124my[39m[38;5;124m'[39m,)); (x, y,) [38;5;241m=[39m Rxy[38;5;241m.[39m_first_ngens([38;5;241m2[39m) File [0;32msrc/cysignals/signals.pyx:310[0m, in [0;36mcysignals.signals.python_check_interrupt[0;34m()[0m [0;31mKeyboardInterrupt[0m:
Rt.<t> = LaurentSeriesRing(GF(p))p = 5
m = 1
Rx.<x> = PolynomialRing(GF(p))
f = x
C_super = superelliptic(f, m)
print(C_super.genus())0
Rxy.<x, y> = PolynomialRing(GF(p), 2)
f1 = superelliptic_function(C_super, x^3)
AS = as_cover(C_super, [f1], prec=200)AS.jumps[[3]]
AS.holomorphic_differentials_basis()[(1) * dx, (z0) * dx, (z0^2) * dx, (x) * dx]
len(AS.holomorphic_differentials_basis())9
p = 3
m = 2
Rx.<x> = PolynomialRing(GF(p))
f = x^m + 1
C_super = superelliptic(f, m)
Rxy.<x, y> = PolynomialRing(GF(p), 2)
f1 = superelliptic_function(C_super, x)
f2 = superelliptic_function(C_super, x^2)
AS = as_cover(C_super, [f1, f2], prec=300)
print(AS.jumps)
print(AS.holomorphic_differentials_basis())[[1, 4], [1, 4]] [(1) * dx, (-z0^2 + z1) * dx, (z0) * dx, (1/y) * dx, (z1/y) * dx, (z1^2/y) * dx, (z0/y) * dx, (z0*z1/y) * dx, ((-x*z0^2 + z0*z1^2)/y) * dx, (z0^2/y) * dx, (z0^2*z1/y) * dx, (x/y) * dx, ((-x*z0^2 + x*z1)/y) * dx, (x*z0/y) * dx]
AS.genus()14
AS.genus()2
AS.holomorphic_differentials_basis()[(1/y) * dx, (z0/y) * dx, ((-x^2 + z0^2)/y) * dx, ((2*x^3 + 2*x^2*z0 + z0^3)/y) * dx, ((2*x^4 - 2*x^3*z0 - x^2*z0^2 + z0^4)/y) * dx, (x/y) * dx, ((-x^2 + x*z0)/y) * dx, ((x^3 - 2*x^2*z0 + x*z0^2)/y) * dx, ((-x^4 - 2*x^3*z0 + 2*x^2*z0^2 + x*z0^3)/y) * dx]
p = 3
m = 2
Rx.<x> = PolynomialRing(GF(p))
for a in GF(p):
for b in GF(p):
for c in GF(p):
if (a, b, c) != (0, 0, 0):
f = x^2 + 1
C_super = superelliptic(f, m)
print(C_super.genus())
Rxy.<x, y> = PolynomialRing(GF(p), 2)
f1 = superelliptic_function(C_super, x)
f2 = superelliptic_function(C_super, )
try:
AS = as_cover(C_super, [f1, f2], prec=500)
except:
pass
print(a, b, c, AS.jumps)0 no 4 -th root; divide by 2 0 0 1 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 0 0 2 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 0 1 0 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 0 1 1 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 0 1 2 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 0 2 0 [[1, 4], [1, 4]] 0 0 2 1 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 0 2 2 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 1 0 0 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 1 0 1 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 1 0 2 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 1 1 0 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 1 1 1 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 1 1 2 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 1 2 0 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 1 2 1 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 1 2 2 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 2 0 0 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 2 0 1 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 2 0 2 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 2 1 0 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 2 1 1 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 2 1 2 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 2 2 0 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 2 2 1 [[1, 4], [1, 4]] 0 no 4 -th root; divide by 2 2 2 2 [[1, 4], [1, 4]]
A, B = group_action_matrices(AS)
print(A, B)[1 2 1 0 0 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0] [0 1 1 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 1 0 0 1 0 0 1 0 0 0 0] [0 0 0 0 1 0 0 1 0 0 1 0 0 0] [0 0 0 0 0 1 0 0 1 0 0 0 0 0] [0 0 0 0 0 0 1 0 0 2 0 0 0 0] [0 0 0 0 0 0 0 1 0 0 2 0 0 0] [0 0 0 0 0 0 0 0 1 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 0 2 0 0 1 2 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 1 0 0 0 1 1] [1 1 0 0 0 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 1 1 1 0 0 0 0 0 0 0 0] [0 0 0 0 1 2 0 0 0 0 0 0 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 1 1 1 0 0 0 0 0] [0 0 0 0 0 0 0 1 2 0 0 0 0 0] [0 0 0 0 0 0 0 0 1 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 1]
pretty_print(A.jordan_form().str())\(\displaystyle \begin{array}{l}
\verb"[1"\verb" "\verb"1"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0]"\\\\
\verb"[0"\verb" "\verb"1"\verb" "\verb"1|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"1|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0]"\\\\
\verb|[-----+-----+-----+-----+---]|\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|1"\verb" "\verb"1"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"1"\verb" "\verb"1|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"1|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0]"\\\\
\verb|[-----+-----+-----+-----+---]|\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|1"\verb" "\verb"1"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"1"\verb" "\verb"1|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"1|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0]"\\\\
\verb|[-----+-----+-----+-----+---]|\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|1"\verb" "\verb"1"\verb" "\verb"0|0"\verb" "\verb"0]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"1"\verb" "\verb"1|0"\verb" "\verb"0]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"1|0"\verb" "\verb"0]"\\\\
\verb|[-----+-----+-----+-----+---]|\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|1"\verb" "\verb"1]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"1]"
\end{array}\)
pretty_print(B.jordan_form().str())\(\displaystyle \begin{array}{l}
\verb"[1"\verb" "\verb"1"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0|0]"\\\\
\verb"[0"\verb" "\verb"1"\verb" "\verb"1|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0|0]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"1|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0|0]"\\\\
\verb|[-----+-----+---+---+---+-+-]|\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|1"\verb" "\verb"1"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0|0]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"1"\verb" "\verb"1|0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0|0]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"1|0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0|0]"\\\\
\verb|[-----+-----+---+---+---+-+-]|\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|1"\verb" "\verb"1|0"\verb" "\verb"0|0"\verb" "\verb"0|0|0]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"1|0"\verb" "\verb"0|0"\verb" "\verb"0|0|0]"\\\\
\verb|[-----+-----+---+---+---+-+-]|\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0|1"\verb" "\verb"1|0"\verb" "\verb"0|0|0]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"1|0"\verb" "\verb"0|0|0]"\\\\
\verb|[-----+-----+---+---+---+-+-]|\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|1"\verb" "\verb"1|0|0]"\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"1|0|0]"\\\\
\verb|[-----+-----+---+---+---+-+-]|\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|1|0]"\\\\
\verb|[-----+-----+---+---+---+-+-]|\\\\
\verb"[0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0"\verb" "\verb"0|0|1]"
\end{array}\)