def _reduce_cusp(self, c):
r"""
Compute a minimal representative for the given cusp c.
Returns a pair (c', t), where c' is the minimal representative
for the given cusp, and t is either 1 or -1, as explained
below. Largely for internal use.
The minimal representative for a cusp is the element in `P^1(Q)`
in lowest terms with minimal positive denominator, and minimal
positive numerator for that denominator.
Two cusps `u1/v1` and `u2/v2` are equivalent modulo `\Gamma_H(N)`
if and only if
`v1 = h*v2 (mod N)` and `u1 = h^(-1)*u2 (mod gcd(v1,N))`
or
`v1 = -h*v2 (mod N)` and `u1 = -h^(-1)*u2 (mod gcd(v1,N))`
for some `h \in H`. Then t is 1 or -1 as c and c' fall into
the first or second case, respectively.
EXAMPLES::
sage: GammaH(6,[5])._reduce_cusp(Cusp(5,3))
(1/3, -1)
sage: GammaH(12,[5])._reduce_cusp(Cusp(8,9))
(1/3, -1)
sage: GammaH(12,[5])._reduce_cusp(Cusp(5,12))
(Infinity, 1)
sage: GammaH(12,[])._reduce_cusp(Cusp(5,12))
(5/12, 1)
sage: GammaH(21,[5])._reduce_cusp(Cusp(-9/14))
(1/7, 1)
"""
c = Cusp(c)
N = int(self.level())
Cusps = c.parent()
v = int(c.denominator() % N)
H = self._list_of_elements_in_H()
# First, if N | v, take care of this case. If u is in \pm H,
# then we return Infinity. If not, let u_0 be the minimum
# of \{ h*u | h \in \pm H \}. Then return u_0/N.
if not v:
u = c.numerator() % N
if u in H:
return Cusps((1, 0)), 1
if (N - u) in H:
return Cusps((1, 0)), -1
ls = [(u * h) % N for h in H]
m1 = min(ls)
m2 = N - max(ls)
if m1 < m2:
return Cusps((m1, N)), 1
else:
return Cusps((m2, N)), -1
u = int(c.numerator() % v)
gcd = get_gcd(N)
d = gcd(v, N)
# If (N,v) == 1, let v_0 be the minimal element
# in \{ v * h | h \in \pm H \}. Then we either return
# Infinity or 1/v_0, as v is or is not in \pm H,
# respectively.
if d == 1:
if v in H:
return Cusps((0, 1)), 1
if (N - v) in H:
return Cusps((0, 1)), -1
ls = [(v * h) % N for h in H]
m1 = min(ls)
m2 = N - max(ls)
if m1 < m2:
return Cusps((1, m1)), 1
else:
return Cusps((1, m2)), -1
val_min = v
inv_mod = get_inverse_mod(N)
# Now we're in the case (N,v) > 1. So we have to do several
# steps: first, compute v_0 as above. While computing this
# minimum, keep track of *all* pairs of (h,s) which give this
# value of v_0.
hs_ls = [(1, 1)]
for h in H:
tmp = (v * h) % N
if tmp < val_min:
val_min = tmp
hs_ls = [(inv_mod(h, N), 1)]
elif tmp == val_min:
hs_ls.append((inv_mod(h, N), 1))
if (N - tmp) < val_min:
val_min = N - tmp
hs_ls = [(inv_mod(h, N), -1)]
elif (N - tmp) == val_min:
hs_ls.append((inv_mod(h, N), -1))
# Finally, we find our minimal numerator. Let u_1 be the
# minimum of s*h^-1*u mod d as (h,s) ranges over the elements
# of hs_ls. We must find the smallest integer u_0 which is
# smaller than v_0, congruent to u_1 mod d, and coprime to
# v_0. Then u_0/v_0 is our minimal representative.
u_min = val_min
sign = None
for h_inv, s in hs_ls:
tmp = (h_inv * s * u) % d
while gcd(tmp, val_min) > 1 and tmp < u_min:
tmp += d
if tmp < u_min:
u_min = tmp
sign = s
return Cusps((u_min, val_min)), sign
def _reduce_cusp(self, c):
r"""
Compute a minimal representative for the given cusp c.
Returns a pair (c', t), where c' is the minimal representative
for the given cusp, and t is either 1 or -1, as explained
below. Largely for internal use.
The minimal representative for a cusp is the element in `P^1(Q)`
in lowest terms with minimal positive denominator, and minimal
positive numerator for that denominator.
Two cusps `u1/v1` and `u2/v2` are equivalent modulo `\Gamma_H(N)`
if and only if
`v1 = h*v2 (mod N)` and `u1 = h^(-1)*u2 (mod gcd(v1,N))`
or
`v1 = -h*v2 (mod N)` and `u1 = -h^(-1)*u2 (mod gcd(v1,N))`
for some `h \in H`. Then t is 1 or -1 as c and c' fall into
the first or second case, respectively.
EXAMPLES::
sage: GammaH(6,[5])._reduce_cusp(Cusp(5,3))
(1/3, -1)
sage: GammaH(12,[5])._reduce_cusp(Cusp(8,9))
(1/3, -1)
sage: GammaH(12,[5])._reduce_cusp(Cusp(5,12))
(Infinity, 1)
sage: GammaH(12,[])._reduce_cusp(Cusp(5,12))
(5/12, 1)
sage: GammaH(21,[5])._reduce_cusp(Cusp(-9/14))
(1/7, 1)
"""
c = Cusp(c)
N = int(self.level())
Cusps = c.parent()
v = int(c.denominator() % N)
H = self._list_of_elements_in_H()
# First, if N | v, take care of this case. If u is in \pm H,
# then we return Infinity. If not, let u_0 be the minimum
# of \{ h*u | h \in \pm H \}. Then return u_0/N.
if not v:
u = c.numerator() % N
if u in H:
return Cusps((1,0)), 1
if (N-u) in H:
return Cusps((1,0)), -1
ls = [ (u*h)%N for h in H ]
m1 = min(ls)
m2 = N-max(ls)
if m1 < m2:
return Cusps((m1,N)), 1
else:
return Cusps((m2,N)), -1
u = int(c.numerator() % v)
gcd = get_gcd(N)
d = gcd(v,N)
# If (N,v) == 1, let v_0 be the minimal element
# in \{ v * h | h \in \pm H \}. Then we either return
# Infinity or 1/v_0, as v is or is not in \pm H,
# respectively.
if d == 1:
if v in H:
return Cusps((0,1)), 1
if (N-v) in H:
return Cusps((0,1)), -1
ls = [ (v*h)%N for h in H ]
m1 = min(ls)
m2 = N-max(ls)
if m1 < m2:
return Cusps((1,m1)), 1
else:
return Cusps((1,m2)), -1
val_min = v
inv_mod = get_inverse_mod(N)
# Now we're in the case (N,v) > 1. So we have to do several
# steps: first, compute v_0 as above. While computing this
# minimum, keep track of *all* pairs of (h,s) which give this
# value of v_0.
hs_ls = [(1,1)]
for h in H:
tmp = (v*h)%N
if tmp < val_min:
val_min = tmp
hs_ls = [(inv_mod(h,N), 1)]
elif tmp == val_min:
hs_ls.append((inv_mod(h,N), 1))
if (N-tmp) < val_min:
val_min = N - tmp
hs_ls = [(inv_mod(h,N), -1)]
elif (N-tmp) == val_min:
hs_ls.append((inv_mod(h,N), -1))
# Finally, we find our minimal numerator. Let u_1 be the
# minimum of s*h^-1*u mod d as (h,s) ranges over the elements
# of hs_ls. We must find the smallest integer u_0 which is
# smaller than v_0, congruent to u_1 mod d, and coprime to
# v_0. Then u_0/v_0 is our minimal representative.
u_min = val_min
sign = None
for h_inv,s in hs_ls:
tmp = (h_inv * s * u)%d
while gcd(tmp, val_min) > 1 and tmp < u_min:
tmp += d
if tmp < u_min:
u_min = tmp
sign = s
return Cusps((u_min, val_min)), sign
def _coset_reduction_data_first_coord(G):
"""
Compute data used for determining the canonical coset
representative of an element of SL_2(Z) modulo G. This
function specifically returns data needed for the first part
of the reduction step (the first coordinate).
INPUT:
G -- a congruence subgroup Gamma_0(N), Gamma_1(N), or Gamma_H(N).
OUTPUT:
A list v such that
v[u] = (min(u*h: h in H),
gcd(u,N) ,
an h such that h*u = min(u*h: h in H)).
EXAMPLES::
sage: G = Gamma0(12)
sage: sage.modular.arithgroup.congroup_gammaH.GammaH_class._coset_reduction_data_first_coord(G)
[(0, 12, 0), (1, 1, 1), (2, 2, 1), (3, 3, 1), (4, 4, 1), (1, 1, 5), (6, 6, 1),
(1, 1, 7), (4, 4, 5), (3, 3, 7), (2, 2, 5), (1, 1, 11)]
"""
H = [int(x) for x in G._list_of_elements_in_H()]
N = int(G.level())
# Get some useful fast functions for inverse and gcd
inverse_mod = get_inverse_mod(N) # optimal inverse function
gcd = get_gcd(N) # optimal gcd function
# We will be filling this list in below.
reduct_data = [0] * N
# We can fill in 0 and all elements of H immediately
reduct_data[0] = (0, N, 0)
for u in H:
reduct_data[u] = (1, 1, inverse_mod(u, N))
# Make a table of the reduction of H (mod N/d), one for each
# divisor d.
repr_H_mod_N_over_d = {}
for d in divisors(N):
# We special-case N == d because in this case,
# 1 % N_over_d is 0
if N == d:
repr_H_mod_N_over_d[d] = [1]
break
N_over_d = N // d
# For each element of H, we look at its image mod
# N_over_d. If we haven't yet seen it, add it on to
# the end of z.
w = [0] * N_over_d
z = [1]
for x in H:
val = x % N_over_d
if not w[val]:
w[val] = 1
z.append(x)
repr_H_mod_N_over_d[d] = z
# Compute the rest of the tuples. The values left to process
# are those where reduct_data has a 0. Note that several of
# these values are processed on each loop below, so re-index
# each time.
while True:
try:
u = reduct_data.index(0)
except ValueError:
break
d = gcd(u, N)
for x in repr_H_mod_N_over_d[d]:
reduct_data[(u * x) % N] = (u, d, inverse_mod(x, N))
return reduct_data
def _coset_reduction_data_first_coord(G):
"""
Compute data used for determining the canonical coset
representative of an element of SL_2(Z) modulo G. This
function specifically returns data needed for the first part
of the reduction step (the first coordinate).
INPUT:
G -- a congruence subgroup Gamma_0(N), Gamma_1(N), or Gamma_H(N).
OUTPUT:
A list v such that
v[u] = (min(u*h: h in H),
gcd(u,N) ,
an h such that h*u = min(u*h: h in H)).
EXAMPLES::
sage: G = Gamma0(12)
sage: sage.modular.arithgroup.congroup_gammaH.GammaH_class._coset_reduction_data_first_coord(G)
[(0, 12, 0), (1, 1, 1), (2, 2, 1), (3, 3, 1), (4, 4, 1), (1, 1, 5), (6, 6, 1),
(1, 1, 7), (4, 4, 5), (3, 3, 7), (2, 2, 5), (1, 1, 11)]
"""
H = [ int(x) for x in G._list_of_elements_in_H() ]
N = int(G.level())
# Get some useful fast functions for inverse and gcd
inverse_mod = get_inverse_mod(N) # optimal inverse function
gcd = get_gcd(N) # optimal gcd function
# We will be filling this list in below.
reduct_data = [0] * N
# We can fill in 0 and all elements of H immediately
reduct_data[0] = (0,N,0)
for u in H:
reduct_data[u] = (1, 1, inverse_mod(u, N))
# Make a table of the reduction of H (mod N/d), one for each
# divisor d.
repr_H_mod_N_over_d = {}
for d in divisors(N):
# We special-case N == d because in this case,
# 1 % N_over_d is 0
if N == d:
repr_H_mod_N_over_d[d] = [1]
break
N_over_d = N//d
# For each element of H, we look at its image mod
# N_over_d. If we haven't yet seen it, add it on to
# the end of z.
w = [0] * N_over_d
z = [1]
for x in H:
val = x%N_over_d
if not w[val]:
w[val] = 1
z.append(x)
repr_H_mod_N_over_d[d] = z
# Compute the rest of the tuples. The values left to process
# are those where reduct_data has a 0. Note that several of
# these values are processed on each loop below, so re-index
# each time.
while True:
try:
u = reduct_data.index(0)
except ValueError:
break
d = gcd(u, N)
for x in repr_H_mod_N_over_d[d]:
reduct_data[(u*x)%N] = (u, d, inverse_mod(x,N))
return reduct_data