def next_prime_not_dividing(P, I):
while True:
p = P.smallest_integer()
if p == 1:
Q = F.ideal(2)
elif p % 5 in [2,3]: # inert
Q = F.primes_above(next_prime(p))[0]
elif p == 5:
Q = F.ideal(7)
else: # p split
A = F.primes_above(p)
if A[0] == P:
Q = A[1]
else:
Q = F.primes_above(next_prime(p))[0]
if not Q.divides(I):
return Q
else:
P = Q # try again
def next_prime_not_dividing(P, I):
while True:
p = P.smallest_integer()
if p == 1:
Q = F.ideal(2)
elif p % 5 in [2,3]: # inert
Q = F.primes_above(next_prime(p))[0]
elif p == 5:
Q = F.ideal(7)
else: # p split
A = F.primes_above(p)
if A[0] == P:
Q = A[1]
else:
Q = F.primes_above(next_prime(p))[0]
if not Q.divides(I):
return Q
else:
P = Q # try again
def find_mod2_splitting():
"""
Return elements w0, w1, w2, w3 that determine a mod-2 splitting
of the Hamilton quaternion algebra over Q(sqrt(5)).
OUTPUT:
- four 2x2 matrices over the residue field of Q(sqrt(5)) of order 4.
EXAMPLES::
sage: import sage.modular.hilbert.sqrt5_fast_python
sage: w = sage.modular.hilbert.sqrt5_fast_python.find_mod2_splitting(); w
(
[ 1 abar] [ 0 abar + 1] [abar + 1 abar] [1 1]
[abar + 1 0], [ abar 0], [ abar abar + 1], [0 1]
)
sage: w[1]^2, w[2]^2
(
[1 0] [1 0]
[0 1], [0 1]
)
sage: span([vector(z.list()) for z in w]).dimension()
4
"""
P = F.primes_above(2)[0]
k = P.residue_field()
M = MatrixSpace(k,2)
V = k**4
g = k.gen() # image of golden ratio
m1 = M(-1)
sqrt_minus_1 = [(w, V(w.list())) for w in M if w*w == m1]
one = M(1)
v_one = V(one.list())
for w1, v1 in sqrt_minus_1:
for w2, v2 in sqrt_minus_1:
w0 = (g-1)*(w1+w2 - w1*w2)
w3 = g*w0 + w2*w1
if w0*w0 != w0 - 1:
continue
if w3*w3 != -1:
continue
if V.span([w0.list(),v1,v2,w3.list()]).dimension() == 4:
return w0, w1, w2, w3
def find_mod2pow_splitting(i):
P = F.primes_above(2)[0]
if i == 1:
R = ResidueRing(P, 1)
M = MatrixSpace(R, 2)
return [M(matrix_lift(a).list()) for a in find_mod2_splitting()]
R = ResidueRing(P, i)
M = MatrixSpace(R, 2)
# arbitrary lift
wbar = [M(matrix_lift(a).list()) for a in find_mod2pow_splitting(i - 1)]
# Find lifts of wbar[1] and wbar[2] that have square -1
k = P.residue_field()
Mk = MatrixSpace(k, 2)
t = 2**(i - 1)
s = M(t)
L = []
for j in [1, 2]:
C = Mk(matrix_lift(wbar[j]**2 + M(1)) / t)
A = Mk(matrix_lift(wbar[j]))
# Find all matrices B in Mk such that AB+BA=C.
L.append([
wbar[j] + s * M(matrix_lift(B)) for B in Mk if A * B + B * A == C
])
g = M(F.gen())
t = M(t)
two = M(2)
ginv = M(F.gen()**(-1))
for w1 in L[0]:
for w2 in L[1]:
w0 = ginv * (two * g * wbar[3] - w1 - w2 - w1 * w2)
w3 = g * w0 + w2 * w1
if w0 * w0 != w0 - M(1):
continue
if w3 * w3 != M(-1):
continue
return w0, w1, w2, w3
raise ValueError
def find_mod2pow_splitting(i):
P = F.primes_above(2)[0]
if i == 1:
R = ResidueRing(P, 1)
M = MatrixSpace(R, 2)
return [M(matrix_lift(a).list()) for a in find_mod2_splitting()]
R = ResidueRing(P, i)
M = MatrixSpace(R, 2)
# arbitrary lift
wbar = [M(matrix_lift(a).list()) for a in find_mod2pow_splitting(i - 1)]
# Find lifts of wbar[1] and wbar[2] that have square -1
k = P.residue_field()
Mk = MatrixSpace(k, 2)
t = 2 ** (i - 1)
s = M(t)
L = []
for j in [1, 2]:
C = Mk(matrix_lift(wbar[j] ** 2 + M(1)) / t)
A = Mk(matrix_lift(wbar[j]))
# Find all matrices B in Mk such that AB+BA=C.
L.append([wbar[j] + s * M(matrix_lift(B)) for B in Mk if A * B + B * A == C])
g = M(F.gen())
t = M(t)
two = M(2)
ginv = M(F.gen() ** (-1))
for w1 in L[0]:
for w2 in L[1]:
w0 = ginv * (two * g * wbar[3] - w1 - w2 - w1 * w2)
w3 = g * w0 + w2 * w1
if w0 * w0 != w0 - M(1):
continue
if w3 * w3 != M(-1):
continue
return w0, w1, w2, w3
raise ValueError
def find_mod2_splitting():
P = F.primes_above(2)[0]
k = P.residue_field()
M = MatrixSpace(k, 2)
V = k ** 4
g = k.gen() # image of golden ratio
m1 = M(-1)
sqrt_minus_1 = [(w, V(w.list())) for w in M if w * w == m1]
one = M(1)
v_one = V(one.list())
for w1, v1 in sqrt_minus_1:
for w2, v2 in sqrt_minus_1:
w0 = (g - 1) * (w1 + w2 - w1 * w2)
w3 = g * w0 + w2 * w1
if w0 * w0 != w0 - 1:
continue
if w3 * w3 != -1:
continue
if V.span([w0.list(), v1, v2, w3.list()]).dimension() == 4:
return w0, w1, w2, w3
def find_mod2_splitting():
P = F.primes_above(2)[0]
k = P.residue_field()
M = MatrixSpace(k, 2)
V = k**4
g = k.gen() # image of golden ratio
m1 = M(-1)
sqrt_minus_1 = [(w, V(w.list())) for w in M if w * w == m1]
one = M(1)
v_one = V(one.list())
for w1, v1 in sqrt_minus_1:
for w2, v2 in sqrt_minus_1:
w0 = (g - 1) * (w1 + w2 - w1 * w2)
w3 = g * w0 + w2 * w1
if w0 * w0 != w0 - 1:
continue
if w3 * w3 != -1:
continue
if V.span([w0.list(), v1, v2, w3.list()]).dimension() == 4:
return w0, w1, w2, w3
def next_prime_of_characteristic_coprime_to(P, I):
p = next_prime(P.smallest_integer())
N = ZZ(I.norm())
while N%p == 0:
p = next_prime(p)
return F.primes_above(p)[0]
def next_prime_of_characteristic_coprime_to(P, I):
p = next_prime(P.smallest_integer())
N = ZZ(I.norm())
while N%p == 0:
p = next_prime(p)
return F.primes_above(p)[0]
def find_mod2pow_splitting(i):
"""
Return elements w0, w1, w2, w3 that determine a mod-`2^i` splitting
of the Hamilton quaternion algebra over Q(sqrt(5)).
INPUT:
- `i` -- positive integer
OUTPUT:
- four 2x2 matrices over a residue ring of Q(sqrt(5)) of characteristic a power of 2
EXAMPLES::
sage: import sage.modular.hilbert.sqrt5_fast_python
sage: w = sage.modular.hilbert.sqrt5_fast_python.find_mod2pow_splitting(1); w
(
[ 1 g] [ 0 1 + g] [1 + g g] [1 1]
[1 + g 0], [ g 0], [ g 1 + g], [0 1]
)
sage: w = sage.modular.hilbert.sqrt5_fast_python.find_mod2pow_splitting(2); w
(
[ 3 2 + 3*g] [ 0 1 + 3*g] [3 + 3*g 3*g]
[ 1 + g 2], [ g 0], [ 3*g 1 + g],
[3 + 2*g 3 + 2*g]
[ 2 1 + 2*g]
)
sage: w[1]^2
[3 0]
[0 3]
sage: w[2]^2
[3 0]
[0 3]
sage: w = sage.modular.hilbert.sqrt5_fast_python.find_mod2pow_splitting(4)
sage: w[1]^2
[15 0]
[ 0 15]
sage: w[2]^2
[15 0]
[ 0 15]
sage: w[1], w[2]
(
[ 0 1 + 15*g] [15 + 15*g 12 + 7*g]
[ g 0], [ 4 + 11*g 1 + g]
)
TESTS:
The power must be positive::
sage: sage.modular.hilbert.sqrt5_fast_python.find_mod2pow_splitting(0)
Traceback (most recent call last):
...
ValueError: i must be positive
"""
P = F.primes_above(2)[0]
i = ZZ(i)
if i <= 0:
raise ValueError, "i must be positive"
if i == 1:
R = ResidueRing(P, 1)
M = MatrixSpace(R,2)
return tuple([M(matrix_lift(a).list()) for a in find_mod2_splitting()])
R = ResidueRing(P, i)
M = MatrixSpace(R,2)
# arbitrary lift
wbar = [M(matrix_lift(a).list()) for a in find_mod2pow_splitting(i-1)]
# Find lifts of wbar[1] and wbar[2] that have square -1
k = P.residue_field()
Mk = MatrixSpace(k, 2)
t = 2**(i-1)
s = M(t)
L = []
for j in [1,2]:
C = Mk(matrix_lift(wbar[j]**2 + M(1)) / t)
A = Mk(matrix_lift(wbar[j]))
# Find all matrices B in Mk such that AB+BA=C.
L.append([wbar[j]+s*M(matrix_lift(B)) for B in Mk if A*B + B*A == C])
g = M(F.gen())
t = M(t)
two = M(2)
ginv = M(F.gen()**(-1))
for w1 in L[0]:
for w2 in L[1]:
w0 = ginv*(two*g*wbar[3] -w1 -w2 - w1*w2)
w3 = g*w0 + w2*w1
if w0*w0 != w0 - M(1):
continue
if w3*w3 != M(-1):
continue
return w0, w1, w2, w3
raise ValueError
