def quadratic_twist(self, D=None):
"""
Return the quadratic twist of this curve by ``D``.
INPUT:
- ``D`` (default None) the twisting parameter (see below).
In characteristics other than 2, `D` must be nonzero, and the
twist is isomorphic to self after adjoining `\sqrt(D)` to the
base.
In characteristic 2, `D` is arbitrary, and the twist is
isomorphic to self after adjoining a root of `x^2+x+D` to the
base.
In characteristic 2 when `j=0`, this is not implemented.
If the base field `F` is finite, `D` need not be specified,
and the curve returned is the unique curve (up to isomorphism)
defined over `F` isomorphic to the original curve over the
quadratic extension of `F` but not over `F` itself. Over
infinite fields, an error is raised if `D` is not given.
EXAMPLES::
sage: E = EllipticCurve([GF(1103)(1), 0, 0, 107, 340]); E
Elliptic Curve defined by y^2 + x*y = x^3 + 107*x + 340 over Finite Field of size 1103
sage: F=E.quadratic_twist(-1); F
Elliptic Curve defined by y^2 = x^3 + 1102*x^2 + 609*x + 300 over Finite Field of size 1103
sage: E.is_isomorphic(F)
False
sage: E.is_isomorphic(F,GF(1103^2,'a'))
True
A characteristic 2 example::
sage: E=EllipticCurve(GF(2),[1,0,1,1,1])
sage: E1=E.quadratic_twist(1)
sage: E.is_isomorphic(E1)
False
sage: E.is_isomorphic(E1,GF(4,'a'))
True
Over finite fields, the twisting parameter may be omitted::
sage: k.<a> = GF(2^10)
sage: E = EllipticCurve(k,[a^2,a,1,a+1,1])
sage: Et = E.quadratic_twist()
sage: Et # random (only determined up to isomorphism)
Elliptic Curve defined by y^2 + x*y = x^3 + (a^7+a^4+a^3+a^2+a+1)*x^2 + (a^8+a^6+a^4+1) over Finite Field in a of size 2^10
sage: E.is_isomorphic(Et)
False
sage: E.j_invariant()==Et.j_invariant()
True
sage: p=next_prime(10^10)
sage: k = GF(p)
sage: E = EllipticCurve(k,[1,2,3,4,5])
sage: Et = E.quadratic_twist()
sage: Et # random (only determined up to isomorphism)
Elliptic Curve defined by y^2 = x^3 + 7860088097*x^2 + 9495240877*x + 3048660957 over Finite Field of size 10000000019
sage: E.is_isomorphic(Et)
False
sage: k2 = GF(p^2,'a')
sage: E.change_ring(k2).is_isomorphic(Et.change_ring(k2))
True
"""
K=self.base_ring()
char=K.characteristic()
if D is None:
if K.is_finite():
x = rings.polygen(K)
if char==2:
# We find D such that x^2+x+D is irreducible. If the
# degree is odd we can take D=1; otherwise it suffices to
# consider odd powers of a generator.
D = K(1)
if K.degree()%2==0:
D = K.gen()
a = D**2
while len((x**2+x+D).roots())>0:
D *= a
else:
# We could take a multiplicative generator but
# that might be expensive to compute; otherwise
# half the elements will do
D = K.random_element()
while len((x**2-D).roots())>0:
D = K.random_element()
else:
raise ValueError("twisting parameter D must be specified over infinite fields.")
else:
try:
D=K(D)
except ValueError:
raise ValueError("twisting parameter D must be in the base field.")
if char!=2 and D.is_zero():
raise ValueError("twisting parameter D must be nonzero when characteristic is not 2")
if char!=2:
b2,b4,b6,b8=self.b_invariants()
# E is isomorphic to [0,b2,0,8*b4,16*b6]
return EllipticCurve(K,[0,b2*D,0,8*b4*D**2,16*b6*D**3])
# now char==2
if self.j_invariant() !=0: # iff a1!=0
a1,a2,a3,a4,a6=self.ainvs()
E0=self.change_weierstrass_model(a1,a3/a1,0,(a1**2*a4+a3**2)/a1**3)
# which has the form = [1,A2,0,0,A6]
assert E0.a1()==K(1)
assert E0.a3()==K(0)
assert E0.a4()==K(0)
return EllipticCurve(K,[1,E0.a2()+D,0,0,E0.a6()])
else:
raise ValueError("Quadratic twist not implemented in char 2 when j=0")
def characters(self):
r"""
Return the two conjugate characters of `K^\times`, where `K` is some
quadratic extension of `\QQ_p`, defining this representation. This is
fully implemented only in the case where the power of `p` dividing the
level of the form is even, in which case `K` is the unique unramified
quadratic extension of `\QQ_p`.
EXAMPLES:
The first example from _[LW11]::
sage: f = Newform('50a')
sage: Pi = LocalComponent(f, 5)
sage: chars = Pi.characters(); chars
[
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> d, 5 |--> 1,
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> -d - 1, 5 |--> 1
]
sage: chars[0].base_ring()
Number Field in d with defining polynomial x^2 + x + 1
These characters are interchanged by the Frobenius automorphism of `\mathbb{F}_{25}`::
sage: chars[0] == chars[1]**5
True
A more complicated example (higher weight and nontrivial central character)::
sage: f = Newforms(GammaH(25, [6]), 3, names='j')[0]; f
q + j0*q^2 + 1/3*j0^3*q^3 - 1/3*j0^2*q^4 + O(q^6)
sage: Pi = LocalComponent(f, 5)
sage: Pi.characters()
[
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> d, 5 |--> 5,
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> -d - 1/3*j0^3, 5 |--> 5
]
sage: Pi.characters()[0].base_ring()
Number Field in d with defining polynomial x^2 + 1/3*j0^3*x - 1/3*j0^2 over its base field
.. warning::
The above output isn't actually the same as in Example 2 of
_[LW11], due to an error in the published paper (correction
pending) -- the published paper has the inverses of the above
characters.
A higher level example::
sage: f = Newform('81a', names='j'); f
q + j0*q^2 + q^4 - j0*q^5 + O(q^6)
sage: LocalComponent(f, 3).characters() # long time (12s on sage.math, 2012)
[
Character of unramified extension Q_3(s)* (s^2 + 2*s + 2 = 0), of level 2, mapping -2*s |--> -2*d - j0, 4 |--> 1, 3*s + 1 |--> -j0*d - 2, 3 |--> 1,
Character of unramified extension Q_3(s)* (s^2 + 2*s + 2 = 0), of level 2, mapping -2*s |--> 2*d + j0, 4 |--> 1, 3*s + 1 |--> j0*d + 1, 3 |--> 1
]
In the ramified case, it's not fully implemented, and just returns a
string indicating which ramified extension is being considered::
sage: Pi = LocalComponent(Newform('27a'), 3)
sage: Pi.characters()
'Character of Q_3(sqrt(-3))'
sage: Pi = LocalComponent(Newform('54a'), 3)
sage: Pi.characters()
'Character of Q_3(sqrt(3))'
"""
T = self.type_space()
if self.conductor() % 2 == 0:
G = SmoothCharacterGroupUnramifiedQuadratic(
self.prime(), self.coefficient_field())
n = self.conductor() // 2
g = G.quotient_gen(n)
m = g.matrix().change_ring(ZZ).list()
tr = (~T.rho(m)).trace()
# The inverse is needed here because T is the *homological* type space,
# which is dual to the cohomological one that defines the local component.
X = polygen(self.coefficient_field())
theta_poly = X**2 - (-1)**n * tr * X + self.central_character()(
g.norm())
if theta_poly.is_irreducible():
F = self.coefficient_field().extension(theta_poly, "d")
G = G.base_extend(F)
chi1, chi2 = [
G.extend_character(n, self.central_character(), x[0])
for x in theta_poly.roots(G.base_ring())
]
# Consistency checks
assert chi1.restrict_to_Qp() == chi2.restrict_to_Qp(
) == self.central_character()
assert chi1 * chi2 == chi1.parent().compose_with_norm(
self.central_character())
return Sequence([chi1, chi2], check=False, cr=True)
else:
# The ramified case.
p = self.prime()
if p == 2:
# The ramified 2-adic representations aren't classified by admissible pairs. Die.
raise NotImplementedError(
"Computation with ramified 2-adic representations not implemented"
)
if p % 4 == 3:
a = ZZ(-1)
else:
a = ZZ(Zmod(self.prime()).quadratic_nonresidue())
tr1 = (~T.rho([0, 1, a * p, 0])).trace()
tr2 = (~T.rho([0, 1, p, 0])).trace()
if tr1 == tr2 == 0:
# This *can* happen. E.g. if the central character satisfies
# chi(-1) = -1, then we have theta(pi) + theta(-pi) = theta(pi)
# * (1 + -1) = 0. In this case, one can presumably identify
# the character and the extension by some more subtle argument
# but I don't know of a good way to automate the process.
raise NotImplementedError(
"Can't identify ramified quadratic extension -- both traces zero"
)
elif tr1 == 0:
return "Character of Q_%s(sqrt(%s))" % (p, p)
elif tr2 == 0:
return "Character of Q_%s(sqrt(%s))" % (p, a * p)
else:
# At least one of the traces is *always* 0, since the type
# space has to be isomorphic to its twist by the (ramified
# quadratic) character corresponding to the quadratic
# extension.
raise RuntimeError("Can't get here!")
def is_quadratic_twist(self, other):
r"""
Determine whether this curve is a quadratic twist of another.
INPUT:
- ``other`` -- an elliptic curves with the same base field as self.
OUTPUT:
Either 0, if the curves are not quadratic twists, or `D` if
``other`` is ``self.quadratic_twist(D)`` (up to isomorphism).
If ``self`` and ``other`` are isomorphic, returns 1.
If the curves are defined over `\mathbb{Q}`, the output `D` is
a squarefree integer.
.. note::
Not fully implemented in characteristic 2, or in
characteristic 3 when both `j`-invariants are 0.
EXAMPLES::
sage: E = EllipticCurve('11a1')
sage: Et = E.quadratic_twist(-24)
sage: E.is_quadratic_twist(Et)
-6
sage: E1=EllipticCurve([0,0,1,0,0])
sage: E1.j_invariant()
0
sage: E2=EllipticCurve([0,0,0,0,2])
sage: E1.is_quadratic_twist(E2)
2
sage: E1.is_quadratic_twist(E1)
1
sage: type(E1.is_quadratic_twist(E1)) == type(E1.is_quadratic_twist(E2)) #trac 6574
True
::
sage: E1=EllipticCurve([0,0,0,1,0])
sage: E1.j_invariant()
1728
sage: E2=EllipticCurve([0,0,0,2,0])
sage: E1.is_quadratic_twist(E2)
0
sage: E2=EllipticCurve([0,0,0,25,0])
sage: E1.is_quadratic_twist(E2)
5
::
sage: F = GF(101)
sage: E1 = EllipticCurve(F,[4,7])
sage: E2 = E1.quadratic_twist()
sage: D = E1.is_quadratic_twist(E2); D!=0
True
sage: F = GF(101)
sage: E1 = EllipticCurve(F,[4,7])
sage: E2 = E1.quadratic_twist()
sage: D = E1.is_quadratic_twist(E2)
sage: E1.quadratic_twist(D).is_isomorphic(E2)
True
sage: E1.is_isomorphic(E2)
False
sage: F2 = GF(101^2,'a')
sage: E1.change_ring(F2).is_isomorphic(E2.change_ring(F2))
True
A characteristic 3 example::
sage: F = GF(3^5,'a')
sage: E1 = EllipticCurve_from_j(F(1))
sage: E2 = E1.quadratic_twist(-1)
sage: D = E1.is_quadratic_twist(E2); D!=0
True
sage: E1.quadratic_twist(D).is_isomorphic(E2)
True
::
sage: E1 = EllipticCurve_from_j(F(0))
sage: E2 = E1.quadratic_twist()
sage: D = E1.is_quadratic_twist(E2); D
1
sage: E1.is_isomorphic(E2)
True
"""
from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
E = self
F = other
if not is_EllipticCurve(E) or not is_EllipticCurve(F):
raise ValueError("arguments are not elliptic curves")
K = E.base_ring()
zero = K.zero_element()
if not K == F.base_ring():
return zero
j=E.j_invariant()
if j != F.j_invariant():
return zero
if E.is_isomorphic(F):
if K is rings.QQ:
return rings.ZZ(1)
return K.one_element()
char=K.characteristic()
if char==2:
raise NotImplementedError("not implemented in characteristic 2")
elif char==3:
if j==0:
raise NotImplementedError("not implemented in characteristic 3 for curves of j-invariant 0")
D = E.b2()/F.b2()
else:
# now char!=2,3:
c4E,c6E = E.c_invariants()
c4F,c6F = F.c_invariants()
if j==0:
um = c6E/c6F
x=rings.polygen(K)
ulist=(x**3-um).roots(multiplicities=False)
if len(ulist)==0:
D = zero
else:
D = ulist[0]
elif j==1728:
um=c4E/c4F
x=rings.polygen(K)
ulist=(x**2-um).roots(multiplicities=False)
if len(ulist)==0:
D = zero
else:
D = ulist[0]
else:
D = (c6E*c4F)/(c6F*c4E)
# Normalization of output:
if D.is_zero():
return D
if K is rings.QQ:
D = D.squarefree_part()
assert E.quadratic_twist(D).is_isomorphic(F)
return D
def quadratic_twist(self, D=None):
"""
Return the quadratic twist of this curve by ``D``.
INPUT:
- ``D`` (default None) the twisting parameter (see below).
In characteristics other than 2, `D` must be nonzero, and the
twist is isomorphic to self after adjoining `\sqrt(D)` to the
base.
In characteristic 2, `D` is arbitrary, and the twist is
isomorphic to self after adjoining a root of `x^2+x+D` to the
base.
In characteristic 2 when `j=0`, this is not implemented.
If the base field `F` is finite, `D` need not be specified,
and the curve returned is the unique curve (up to isomorphism)
defined over `F` isomorphic to the original curve over the
quadratic extension of `F` but not over `F` itself. Over
infinite fields, an error is raised if `D` is not given.
EXAMPLES::
sage: E = EllipticCurve([GF(1103)(1), 0, 0, 107, 340]); E
Elliptic Curve defined by y^2 + x*y = x^3 + 107*x + 340 over Finite Field of size 1103
sage: F=E.quadratic_twist(-1); F
Elliptic Curve defined by y^2 = x^3 + 1102*x^2 + 609*x + 300 over Finite Field of size 1103
sage: E.is_isomorphic(F)
False
sage: E.is_isomorphic(F,GF(1103^2,'a'))
True
A characteristic 2 example::
sage: E=EllipticCurve(GF(2),[1,0,1,1,1])
sage: E1=E.quadratic_twist(1)
sage: E.is_isomorphic(E1)
False
sage: E.is_isomorphic(E1,GF(4,'a'))
True
Over finite fields, the twisting parameter may be omitted::
sage: k.<a> = GF(2^10)
sage: E = EllipticCurve(k,[a^2,a,1,a+1,1])
sage: Et = E.quadratic_twist()
sage: Et # random (only determined up to isomorphism)
Elliptic Curve defined by y^2 + x*y = x^3 + (a^7+a^4+a^3+a^2+a+1)*x^2 + (a^8+a^6+a^4+1) over Finite Field in a of size 2^10
sage: E.is_isomorphic(Et)
False
sage: E.j_invariant()==Et.j_invariant()
True
sage: p=next_prime(10^10)
sage: k = GF(p)
sage: E = EllipticCurve(k,[1,2,3,4,5])
sage: Et = E.quadratic_twist()
sage: Et # random (only determined up to isomorphism)
Elliptic Curve defined by y^2 = x^3 + 7860088097*x^2 + 9495240877*x + 3048660957 over Finite Field of size 10000000019
sage: E.is_isomorphic(Et)
False
sage: k2 = GF(p^2,'a')
sage: E.change_ring(k2).is_isomorphic(Et.change_ring(k2))
True
"""
K = self.base_ring()
char = K.characteristic()
if D is None:
if K.is_finite():
x = rings.polygen(K)
if char == 2:
# We find D such that x^2+x+D is irreducible. If the
# degree is odd we can take D=1; otherwise it suffices to
# consider odd powers of a generator.
D = K(1)
if K.degree() % 2 == 0:
D = K.gen()
a = D**2
while len((x**2 + x + D).roots()) > 0:
D *= a
else:
# We could take a multiplicative generator but
# that might be expensive to compute; otherwise
# half the elements will do
D = K.random_element()
while len((x**2 - D).roots()) > 0:
D = K.random_element()
else:
raise ValueError(
"twisting parameter D must be specified over infinite fields."
)
else:
try:
D = K(D)
except ValueError:
raise ValueError(
"twisting parameter D must be in the base field.")
if char != 2 and D.is_zero():
raise ValueError(
"twisting parameter D must be nonzero when characteristic is not 2"
)
if char != 2:
b2, b4, b6, b8 = self.b_invariants()
# E is isomorphic to [0,b2,0,8*b4,16*b6]
return EllipticCurve(K,
[0, b2 * D, 0, 8 * b4 * D**2, 16 * b6 * D**3])
# now char==2
if self.j_invariant() != 0: # iff a1!=0
a1, a2, a3, a4, a6 = self.ainvs()
E0 = self.change_weierstrass_model(a1, a3 / a1, 0,
(a1**2 * a4 + a3**2) / a1**3)
# which has the form = [1,A2,0,0,A6]
assert E0.a1() == K(1)
assert E0.a3() == K(0)
assert E0.a4() == K(0)
return EllipticCurve(K, [1, E0.a2() + D, 0, 0, E0.a6()])
else:
raise ValueError(
"Quadratic twist not implemented in char 2 when j=0")
def characters(self):
r"""
Return the two conjugate characters of `K^\times`, where `K` is some
quadratic extension of `\QQ_p`, defining this representation. This is
fully implemented only in the case where the power of `p` dividing the
level of the form is even, in which case `K` is the unique unramified
quadratic extension of `\QQ_p`.
EXAMPLES:
The first example from _[LW11]::
sage: f = Newform('50a')
sage: Pi = LocalComponent(f, 5)
sage: chars = Pi.characters(); chars
[
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> d, 5 |--> 1,
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> -d - 1, 5 |--> 1
]
sage: chars[0].base_ring()
Number Field in d with defining polynomial x^2 + x + 1
These characters are interchanged by the Frobenius automorphism of `\mathbb{F}_{25}`::
sage: chars[0] == chars[1]**5
True
A more complicated example (higher weight and nontrivial central character)::
sage: f = Newforms(GammaH(25, [6]), 3, names='j')[0]; f
q + j0*q^2 + 1/3*j0^3*q^3 - 1/3*j0^2*q^4 + O(q^6)
sage: Pi = LocalComponent(f, 5)
sage: Pi.characters()
[
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> d, 5 |--> 5,
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> -d - 1/3*j0^3, 5 |--> 5
]
sage: Pi.characters()[0].base_ring()
Number Field in d with defining polynomial x^2 + 1/3*j0^3*x - 1/3*j0^2 over its base field
.. warning::
The above output isn't actually the same as in Example 2 of
_[LW11], due to an error in the published paper (correction
pending) -- the published paper has the inverses of the above
characters.
A higher level example::
sage: f = Newform('81a', names='j'); f
q + j0*q^2 + q^4 - j0*q^5 + O(q^6)
sage: LocalComponent(f, 3).characters()
[
Character of unramified extension Q_3(s)* (s^2 + 2*s + 2 = 0), of level 2, mapping -2*s |--> -2*d - j0, 4 |--> 1, 3*s + 1 |--> -j0*d - 2, 3 |--> 1,
Character of unramified extension Q_3(s)* (s^2 + 2*s + 2 = 0), of level 2, mapping -2*s |--> 2*d + j0, 4 |--> 1, 3*s + 1 |--> j0*d + 1, 3 |--> 1
]
In the ramified case, it's not fully implemented, and just returns a
string indicating which ramified extension is being considered::
sage: Pi = LocalComponent(Newform('27a'), 3)
sage: Pi.characters()
'Character of Q_3(sqrt(-3))'
sage: Pi = LocalComponent(Newform('54a'), 3)
sage: Pi.characters()
'Character of Q_3(sqrt(3))'
"""
T = self.type_space()
if self.conductor() % 2 == 0:
G = SmoothCharacterGroupUnramifiedQuadratic(self.prime(), self.coefficient_field())
n = self.conductor() // 2
g = G.quotient_gen(n)
m = g.matrix().change_ring(ZZ).list()
tr = (~T.rho(m)).trace()
# The inverse is needed here because T is the *homological* type space,
# which is dual to the cohomological one that defines the local component.
X = polygen(self.coefficient_field())
theta_poly = X**2 - (-1)**n*tr*X + self.central_character()(g.norm())
if theta_poly.is_irreducible():
F = self.coefficient_field().extension(theta_poly, "d")
G = G.base_extend(F)
chi1, chi2 = [G.extend_character(n, self.central_character(), x[0]) for x in theta_poly.roots(G.base_ring())]
# Consistency checks
assert chi1.restrict_to_Qp() == chi2.restrict_to_Qp() == self.central_character()
assert chi1*chi2 == chi1.parent().compose_with_norm(self.central_character())
return Sequence([chi1, chi2], check=False, cr=True)
else:
# The ramified case.
p = self.prime()
if p == 2:
# The ramified 2-adic representations aren't classified by admissible pairs. Die.
raise NotImplementedError( "Computation with ramified 2-adic representations not implemented" )
if p % 4 == 3:
a = ZZ(-1)
else:
a = ZZ(Zmod(self.prime()).quadratic_nonresidue())
tr1 = (~T.rho([0,1,a*p, 0])).trace()
tr2 = (~T.rho([0,1,p,0])).trace()
if tr1 == tr2 == 0:
# This *can* happen. E.g. if the central character satisfies
# chi(-1) = -1, then we have theta(pi) + theta(-pi) = theta(pi)
# * (1 + -1) = 0. In this case, one can presumably identify
# the character and the extension by some more subtle argument
# but I don't know of a good way to automate the process.
raise NotImplementedError( "Can't identify ramified quadratic extension -- both traces zero" )
elif tr1 == 0:
return "Character of Q_%s(sqrt(%s))" % (p, p)
elif tr2 == 0:
return "Character of Q_%s(sqrt(%s))" % (p, a*p)
else:
# At least one of the traces is *always* 0, since the type
# space has to be isomorphic to its twist by the (ramified
# quadratic) character corresponding to the quadratic
# extension.
raise RuntimeError( "Can't get here!" )
def is_quadratic_twist(self, other):
r"""
Determine whether this curve is a quadratic twist of another.
INPUT:
- ``other`` -- an elliptic curves with the same base field as self.
OUTPUT:
Either 0, if the curves are not quadratic twists, or `D` if
``other`` is ``self.quadratic_twist(D)`` (up to isomorphism).
If ``self`` and ``other`` are isomorphic, returns 1.
If the curves are defined over `\mathbb{Q}`, the output `D` is
a squarefree integer.
.. note::
Not fully implemented in characteristic 2, or in
characteristic 3 when both `j`-invariants are 0.
EXAMPLES::
sage: E = EllipticCurve('11a1')
sage: Et = E.quadratic_twist(-24)
sage: E.is_quadratic_twist(Et)
-6
sage: E1=EllipticCurve([0,0,1,0,0])
sage: E1.j_invariant()
0
sage: E2=EllipticCurve([0,0,0,0,2])
sage: E1.is_quadratic_twist(E2)
2
sage: E1.is_quadratic_twist(E1)
1
sage: type(E1.is_quadratic_twist(E1)) == type(E1.is_quadratic_twist(E2)) #trac 6574
True
::
sage: E1=EllipticCurve([0,0,0,1,0])
sage: E1.j_invariant()
1728
sage: E2=EllipticCurve([0,0,0,2,0])
sage: E1.is_quadratic_twist(E2)
0
sage: E2=EllipticCurve([0,0,0,25,0])
sage: E1.is_quadratic_twist(E2)
5
::
sage: F = GF(101)
sage: E1 = EllipticCurve(F,[4,7])
sage: E2 = E1.quadratic_twist()
sage: D = E1.is_quadratic_twist(E2); D!=0
True
sage: F = GF(101)
sage: E1 = EllipticCurve(F,[4,7])
sage: E2 = E1.quadratic_twist()
sage: D = E1.is_quadratic_twist(E2)
sage: E1.quadratic_twist(D).is_isomorphic(E2)
True
sage: E1.is_isomorphic(E2)
False
sage: F2 = GF(101^2,'a')
sage: E1.change_ring(F2).is_isomorphic(E2.change_ring(F2))
True
A characteristic 3 example::
sage: F = GF(3^5,'a')
sage: E1 = EllipticCurve_from_j(F(1))
sage: E2 = E1.quadratic_twist(-1)
sage: D = E1.is_quadratic_twist(E2); D!=0
True
sage: E1.quadratic_twist(D).is_isomorphic(E2)
True
::
sage: E1 = EllipticCurve_from_j(F(0))
sage: E2 = E1.quadratic_twist()
sage: D = E1.is_quadratic_twist(E2); D
1
sage: E1.is_isomorphic(E2)
True
"""
from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
E = self
F = other
if not is_EllipticCurve(E) or not is_EllipticCurve(F):
raise ValueError("arguments are not elliptic curves")
K = E.base_ring()
zero = K.zero()
if not K == F.base_ring():
return zero
j = E.j_invariant()
if j != F.j_invariant():
return zero
if E.is_isomorphic(F):
if K is rings.QQ:
return rings.ZZ(1)
return K.one()
char = K.characteristic()
if char == 2:
raise NotImplementedError("not implemented in characteristic 2")
elif char == 3:
if j == 0:
raise NotImplementedError(
"not implemented in characteristic 3 for curves of j-invariant 0"
)
D = E.b2() / F.b2()
else:
# now char!=2,3:
c4E, c6E = E.c_invariants()
c4F, c6F = F.c_invariants()
if j == 0:
um = c6E / c6F
x = rings.polygen(K)
ulist = (x**3 - um).roots(multiplicities=False)
if len(ulist) == 0:
D = zero
else:
D = ulist[0]
elif j == 1728:
um = c4E / c4F
x = rings.polygen(K)
ulist = (x**2 - um).roots(multiplicities=False)
if len(ulist) == 0:
D = zero
else:
D = ulist[0]
else:
D = (c6E * c4F) / (c6F * c4E)
# Normalization of output:
if D.is_zero():
return D
if K is rings.QQ:
D = D.squarefree_part()
assert E.quadratic_twist(D).is_isomorphic(F)
return D
def characters(self):
r"""
Return the two conjugate characters of `K^\times`, where `K` is some
quadratic extension of `\QQ_p`, defining this representation. An error
will be raised in some 2-adic cases, since not all 2-adic supercuspidal
representations arise in this way.
EXAMPLES:
The first example from [LW2012]_::
sage: f = Newform('50a')
sage: Pi = LocalComponent(f, 5)
sage: chars = Pi.characters(); chars
[
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> -d - 1, 5 |--> 1,
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> d, 5 |--> 1
]
sage: chars[0].base_ring()
Number Field in d with defining polynomial x^2 + x + 1
These characters are interchanged by the Frobenius automorphism of `\GF{25}`::
sage: chars[0] == chars[1]**5
True
A more complicated example (higher weight and nontrivial central character)::
sage: f = Newforms(GammaH(25, [6]), 3, names='j')[0]; f
q + j0*q^2 + 1/3*j0^3*q^3 - 1/3*j0^2*q^4 + O(q^6)
sage: Pi = LocalComponent(f, 5)
sage: Pi.characters()
[
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> 1/3*j0^2*d - 1/3*j0^3, 5 |--> 5,
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> -1/3*j0^2*d, 5 |--> 5
]
sage: Pi.characters()[0].base_ring()
Number Field in d with defining polynomial x^2 - j0*x + 1/3*j0^2 over its base field
.. warning::
The above output isn't actually the same as in Example 2 of
[LW2012]_, due to an error in the published paper (correction
pending) -- the published paper has the inverses of the above
characters.
A higher level example::
sage: f = Newform('81a', names='j'); f
q + j0*q^2 + q^4 - j0*q^5 + O(q^6)
sage: LocalComponent(f, 3).characters() # long time (12s on sage.math, 2012)
[
Character of unramified extension Q_3(s)* (s^2 + 2*s + 2 = 0), of level 2, mapping -2*s |--> -2*d + j0, 4 |--> 1, 3*s + 1 |--> -j0*d + 1, 3 |--> 1,
Character of unramified extension Q_3(s)* (s^2 + 2*s + 2 = 0), of level 2, mapping -2*s |--> 2*d - j0, 4 |--> 1, 3*s + 1 |--> j0*d - 2, 3 |--> 1
]
Some ramified examples::
sage: Newform('27a').local_component(3).characters()
[
Character of ramified extension Q_3(s)* (s^2 - 6 = 0), of level 2, mapping 2 |--> 1, s + 1 |--> -d, s |--> -1,
Character of ramified extension Q_3(s)* (s^2 - 6 = 0), of level 2, mapping 2 |--> 1, s + 1 |--> d - 1, s |--> -1
]
sage: LocalComponent(Newform('54a'), 3, twist_factor=4).characters()
[
Character of ramified extension Q_3(s)* (s^2 - 3 = 0), of level 2, mapping 2 |--> 1, s + 1 |--> -1/9*d, s |--> -9,
Character of ramified extension Q_3(s)* (s^2 - 3 = 0), of level 2, mapping 2 |--> 1, s + 1 |--> 1/9*d - 1, s |--> -9
]
A 2-adic non-example::
sage: Newform('24a').local_component(2).characters()
Traceback (most recent call last):
...
ValueError: Totally ramified 2-adic representations are not classified by characters
Examples where `K^\times / \QQ_p^\times` is not topologically cyclic
(which complicates the computations greatly)::
sage: Newforms(DirichletGroup(64, QQ).1, 2, names='a')[0].local_component(2).characters() # long time, random
[
Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 3, mapping s |--> 1, 2*s + 1 |--> 1/2*a0, 4*s + 1 |--> 1, -1 |--> 1, 2 |--> 1,
Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 3, mapping s |--> 1, 2*s + 1 |--> 1/2*a0, 4*s + 1 |--> -1, -1 |--> 1, 2 |--> 1
]
sage: Newform('243a',names='a').local_component(3).characters() # long time
[
Character of ramified extension Q_3(s)* (s^2 - 6 = 0), of level 4, mapping -2*s - 1 |--> -d - 1, 4 |--> 1, 3*s + 1 |--> -d - 1, s |--> 1,
Character of ramified extension Q_3(s)* (s^2 - 6 = 0), of level 4, mapping -2*s - 1 |--> d, 4 |--> 1, 3*s + 1 |--> d, s |--> 1
]
"""
T = self.type_space()
p = self.prime()
if self.conductor() % 2 == 0:
G = SmoothCharacterGroupUnramifiedQuadratic(
self.prime(), self.coefficient_field())
n = self.conductor() // 2
gs = G.quotient_gens(n)
g = gs[-1]
assert g.valuation(G.ideal(1)) == 0
m = g.matrix().change_ring(ZZ).list()
tr = (~T.rho(m)).trace()
# The inverse is needed here because T is the *homological* type space,
# which is dual to the cohomological one that defines the local component.
X = polygen(self.coefficient_field())
theta_poly = X**2 - (-1)**n * tr * X + self.central_character()(
g.norm())
verbose("theta_poly for %s is %s" % (g, theta_poly), level=1)
if theta_poly.is_irreducible():
F = self.coefficient_field().extension(theta_poly, "d")
G = G.base_extend(F)
# roots with repetitions allowed
gvals = flatten([[y[0]] * y[1]
for y in theta_poly.roots(G.base_ring())])
if len(gs) == 1:
# This is always the case if p != 2
chi1, chi2 = [
G.extend_character(n, self.central_character(), [x])
for x in gvals
]
else:
# 2-adic cases, conductor >= 64. Here life is complicated
# because the quotient (O_K* / p^n)^* / (image of Z_2^*) is not
# cyclic.
g0 = gs[0]
try:
G._reduce_Qp(1, g0)
raise ArithmeticError("Bad generators returned")
except ValueError:
pass
tr = (~T.rho(g0.matrix().list())).trace()
X = polygen(G.base_ring())
theta0_poly = X**2 - (
-1)**n * tr * X + self.central_character()(g0.norm())
verbose("theta_poly for %s is %s" % (g0, theta_poly), level=1)
if theta0_poly.is_irreducible():
F = theta0_poly.base_ring().extension(theta_poly, "e")
G = G.base_extend(F)
g0vals = flatten([[y[0]] * y[1]
for y in theta0_poly.roots(G.base_ring())])
pairA = [[g0vals[0], gvals[0]], [g0vals[1], gvals[1]]]
pairB = [[g0vals[0], gvals[1]], [g0vals[1], gvals[0]]]
A_fail = 0
B_fail = 0
try:
chisA = [
G.extend_character(n, self.central_character(), [y, x])
for (y, x) in pairA
]
except ValueError:
A_fail = 1
try:
chisB = [
G.extend_character(n, self.central_character(), [y, x])
for (y, x) in pairB
]
except ValueError:
B_fail = 1
if chisA == chisB or chisA == reversed(chisB):
# repeated roots -- break symmetry arbitrarily
B_fail = 1
# check the character relation from LW12
if (not A_fail and not B_fail):
for x in G.ideal(n).invertible_residues():
try:
# test if G mod p is in Fp
flag = G._reduce_Qp(1, x)
except ValueError:
flag = None
if flag is not None:
verbose("skipping x=%s as congruent to %s mod p" %
(x, flag))
continue
verbose("testing x = %s" % x, level=1)
ti = (-1)**n * (~T.rho(x.matrix().list())).trace()
verbose(" trace of matrix is %s" % ti, level=1)
if ti != chisA[0](x) + chisA[1](x):
verbose(" chisA FAILED", level=1)
A_fail = 1
break
if ti != chisB[0](x) + chisB[1](x):
verbose(" chisB FAILED", level=1)
B_fail = 1
break
else:
verbose(" Trace identity check works for both",
level=1)
if B_fail and not A_fail:
chi1, chi2 = chisA
elif A_fail and not B_fail:
chi1, chi2 = chisB
else:
raise ValueError(
"Something went wrong: can't identify the characters")
# Consistency checks
assert chi1.restrict_to_Qp() == chi2.restrict_to_Qp(
) == self.central_character()
assert chi1 * chi2 == chi1.parent().compose_with_norm(
self.central_character())
return Sequence([chi1, chi2], check=False, cr=True)
else:
# The ramified case.
n = self.conductor() - 1
if p == 2:
# The ramified 2-adic representations aren't classified by admissible pairs. Die.
raise ValueError(
"Totally ramified 2-adic representations are not classified by characters"
)
G0 = SmoothCharacterGroupRamifiedQuadratic(
p, 0, self.coefficient_field())
G1 = SmoothCharacterGroupRamifiedQuadratic(
p, 1, self.coefficient_field())
q0 = G0.quotient_gens(n)
assert all(x.valuation(G0.ideal(1)) == 1 for x in q0)
q1 = G1.quotient_gens(n)
assert all(x.valuation(G1.ideal(1)) == 1 for x in q1)
t0 = [(~T.rho(q.matrix().list())).trace() for q in q0]
t1 = [(~T.rho(q.matrix().list())).trace() for q in q1]
if all(x == 0 for x in t0 + t1):
# Can't happen?
raise NotImplementedError(
"Can't identify ramified quadratic extension -- all traces zero"
)
elif all(x == 0 for x in t1):
G, qs, ts = G0, q0, t0
elif all(x == 0 for x in t0):
G, qs, ts = G1, q1, t1
else:
# At least one of the traces is *always* 0, since the type
# space has to be isomorphic to its twist by the (ramified
# quadratic) character corresponding to the quadratic
# extension.
raise RuntimeError("Can't get here!")
q = qs[0]
t = ts[0]
k = self.newform().weight()
t *= p**ZZ((k - 2 + self.twist_factor()) / 2)
X = polygen(self.coefficient_field())
theta_poly = X**2 - X * t + self.central_character()(q.norm())
verbose("theta_poly is %s" % theta_poly, level=1)
if theta_poly.is_irreducible():
F = self.coefficient_field().extension(theta_poly, "d")
G = G.base_extend(F)
c1q, c2q = flatten([[x] * e
for x, e in theta_poly.roots(G.base_ring())])
if len(qs) == 1:
chi1, chi2 = [
G.extend_character(n, self.central_character(), [x])
for x in [c1q, c2q]
]
else:
assert p == 3
q = qs[1]
t = ts[1]
t *= p**ZZ((k - 2 + self.twist_factor()) / 2)
X = polygen(G.base_ring())
theta_poly = X**2 - X * t + self.central_character()(q.norm())
verbose("theta_poly is %s" % theta_poly, level=1)
if theta_poly.is_irreducible():
F = G.base_ring().extension(theta_poly, "e")
G = G.base_extend(F)
c1q2, c2q2 = flatten(
[[x] * e for x, e in theta_poly.roots(G.base_ring())])
pairA = [[c1q, c1q2], [c2q, c2q2]]
pairB = [[c1q, c2q2], [c2q, c1q2]]
A_fail = 0
B_fail = 0
try:
chisA = [
G.extend_character(n, self.central_character(), [x, y])
for (x, y) in pairA
]
except ValueError:
verbose('A failed to create', level=1)
A_fail = 1
try:
chisB = [
G.extend_character(n, self.central_character(), [x, y])
for (x, y) in pairB
]
except ValueError:
verbose('A failed to create', level=1)
B_fail = 1
if c1q == c2q or c1q2 == c2q2:
B_fail = 1
for u in G.ideal(n).invertible_residues():
if A_fail or B_fail:
break
x = q * u
verbose("testing x = %s" % x, level=1)
ti = (~T.rho(x.matrix().list())).trace() * p**ZZ(
(k - 2 + self.twist_factor()) / 2)
verbose("trace of matrix is %s" % ti, level=1)
if chisA[0](x) + chisA[1](x) != ti:
A_fail = 1
if chisB[0](x) + chisB[1](x) != ti:
B_fail = 1
if B_fail and not A_fail:
chi1, chi2 = chisA
elif A_fail and not B_fail:
chi1, chi2 = chisB
else:
raise ValueError(
"Something went wrong: can't identify the characters")
# Consistency checks
assert chi1.restrict_to_Qp() == chi2.restrict_to_Qp(
) == self.central_character()
assert chi1 * chi2 == chi1.parent().compose_with_norm(
self.central_character())
return Sequence([chi1, chi2], check=False, cr=True)
def mcmullen_genus2_prototype(w, h, t, e, rel=0):
r"""
McMullen prototypes in the stratum H(2).
These prototype appear at least in McMullen "Teichmüller curves in genus
two: Discriminant and spin" (2004). The notation from that paper are
quadruple ``(a, b, c, e)`` which translates in our notation as
``w = b``, ``h = c``, ``t = a`` (and ``e = e``).
The associated discriminant is `D = e^2 + 4 wh`.
If ``rel`` is a positive parameter (less than w-lambda) the surface belongs
to the eigenform locus in H(1,1).
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from surface_dynamics import AbelianStratum
sage: prototypes = {
....: 5: [(1,1,0,-1)],
....: 8: [(1,1,0,-2), (2,1,0,0)],
....: 9: [(2,1,0,-1)],
....: 12: [(1,2,0,-2), (2,1,0,-2), (3,1,0,0)],
....: 13: [(1,1,0,-3), (3,1,0,-1), (3,1,0,1)],
....: 16: [(3,1,0,-2), (4,1,0,0)],
....: 17: [(1,2,0,-3), (2,1,0,-3), (2,2,0,-1), (2,2,1,-1), (4,1,0,-1), (4,1,0,1)],
....: 20: [(1,1,0,-4), (2,2,1,-2), (4,1,0,-2), (4,1,0,2)],
....: 21: [(1,3,0,-3), (3,1,0,-3)],
....: 24: [(1,2,0,-4), (2,1,0,-4), (3,2,0,0)],
....: 25: [(2,2,0,-3), (2,2,1,-3), (3,2,0,-1), (4,1,0,-3)]}
sage: for D in sorted(prototypes):
....: for w,h,t,e in prototypes[D]:
....: T = translation_surfaces.mcmullen_genus2_prototype(w,h,t,e)
....: assert T.stratum() == AbelianStratum(2)
....: assert (D.is_square() and T.base_ring() is QQ) or (T.base_ring().polynomial().discriminant() == D)
An example with some relative homology::
sage: U8 = translation_surfaces.mcmullen_genus2_prototype(2,1,0,0,1/4) # discriminant 8
sage: U12 = translation_surfaces.mcmullen_genus2_prototype(3,1,0,0,3/10) # discriminant 12
sage: U8.stratum()
H_2(1^2)
sage: U8.base_ring().polynomial().discriminant()
8
sage: U8.j_invariant()
(
[4 0]
(0), (0), [0 2]
)
sage: U12.stratum()
H_2(1^2)
sage: U12.base_ring().polynomial().discriminant()
12
sage: U12.j_invariant()
(
[6 0]
(0), (0), [0 2]
)
"""
w = ZZ(w)
h = ZZ(h)
t = ZZ(t)
e = ZZ(e)
g = w.gcd(h)
gg = g.gcd(t).gcd(e)
if w <= 0 or h <= 0 or t < 0 or t >= g or not g.gcd(t).gcd(
e).is_one() or e + h >= w:
raise ValueError("invalid parameters")
x = polygen(QQ)
poly = x**2 - e * x - w * h
if poly.is_irreducible():
emb = AA.polynomial_root(poly, RIF(0, w))
K = NumberField(poly, 'l', embedding=emb)
l = K.gen()
else:
K = QQ
D = e**2 + 4 * w * h
d = D.sqrt()
l = (e + d) / 2
rel = K(rel)
# (lambda,lambda) square on top
# twisted (w,0), (t,h)
s = Surface_list(base_ring=K)
if rel:
if rel < 0 or rel > w - l:
raise ValueError("invalid rel argument")
s.add_polygon(
polygons(vertices=[(0, 0), (l, 0), (l + rel, l), (rel, l)],
ring=K))
s.add_polygon(
polygons(vertices=[(0, 0), (rel, 0), (rel + l, 0), (w, 0),
(w + t, h), (l + rel + t, h), (t + l, h),
(t, h)],
ring=K))
s.set_edge_pairing(0, 1, 0, 3)
s.set_edge_pairing(0, 0, 1, 6)
s.set_edge_pairing(0, 2, 1, 1)
s.set_edge_pairing(1, 2, 1, 4)
s.set_edge_pairing(1, 3, 1, 7)
s.set_edge_pairing(1, 0, 1, 5)
else:
s.add_polygon(
polygons(vertices=[(0, 0), (l, 0), (l, l), (0, l)], ring=K))
s.add_polygon(
polygons(vertices=[(0, 0), (l, 0), (w, 0), (w + t, h),
(l + t, h), (t, h)],
ring=K))
s.set_edge_pairing(0, 1, 0, 3)
s.set_edge_pairing(0, 0, 1, 4)
s.set_edge_pairing(0, 2, 1, 0)
s.set_edge_pairing(1, 1, 1, 3)
s.set_edge_pairing(1, 2, 1, 5)
s.set_immutable()
return TranslationSurface(s)
def arnoux_yoccoz(genus):
r"""
Construct the Arnoux-Yoccoz surface of genus 3 or greater.
This presentation of the surface follows Section 2.3 of
Joshua P. Bowman's paper "The Complete Family of Arnoux-Yoccoz
Surfaces."
EXAMPLES::
sage: from flatsurf import *
sage: s = translation_surfaces.arnoux_yoccoz(4)
sage: TestSuite(s).run()
sage: s.is_delaunay_decomposed()
True
sage: s = s.canonicalize()
sage: field=s.base_ring()
sage: a = field.gen()
sage: from sage.matrix.constructor import Matrix
sage: m = Matrix([[a,0],[0,~a]])
sage: ss = m*s
sage: ss = ss.canonicalize()
sage: s.cmp(ss) == 0
True
The Arnoux-Yoccoz pseudo-Anosov are known to have (minimal) invariant
foliations with SAF=0::
sage: S3 = translation_surfaces.arnoux_yoccoz(3)
sage: Jxx, Jyy, Jxy = S3.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0 2 0]
[ 2 -2 0]
[ 0 0 2]
sage: S4 = translation_surfaces.arnoux_yoccoz(4)
sage: Jxx, Jyy, Jxy = S4.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0 2 0 0]
[ 2 -2 0 0]
[ 0 0 2 2]
[ 0 0 2 0]
"""
g = ZZ(genus)
assert g >= 3
x = polygen(AA)
p = sum([x**i for i in range(1, g + 1)]) - 1
cp = AA.common_polynomial(p)
alpha_AA = AA.polynomial_root(cp, RIF(1 / 2, 1))
field = NumberField(alpha_AA.minpoly(), 'alpha', embedding=alpha_AA)
a = field.gen()
V = VectorSpace(field, 2)
p = [None for i in range(g + 1)]
q = [None for i in range(g + 1)]
p[0] = V(((1 - a**g) / 2, a**2 / (1 - a)))
q[0] = V((-a**g / 2, a))
p[1] = V((-(a**(g - 1) + a**g) / 2, (a - a**2 + a**3) / (1 - a)))
p[g] = V((1 + (a - a**g) / 2, (3 * a - 1 - a**2) / (1 - a)))
for i in range(2, g):
p[i] = V(((a - a**i) / (1 - a), a / (1 - a)))
for i in range(1, g + 1):
q[i] = V(((2 * a - a**i - a**(i + 1)) / (2 * (1 - a)),
(a - a**(g - i + 2)) / (1 - a)))
P = ConvexPolygons(field)
s = Surface_list(field)
T = [None] * (2 * g + 1)
Tp = [None] * (2 * g + 1)
from sage.matrix.constructor import Matrix
m = Matrix([[1, 0], [0, -1]])
for i in range(1, g + 1):
# T_i is (P_0,Q_i,Q_{i-1})
T[i] = s.add_polygon(
P(edges=[q[i] - p[0], q[i - 1] - q[i], p[0] - q[i - 1]]))
# T_{g+i} is (P_i,Q_{i-1},Q_{i})
T[g + i] = s.add_polygon(
P(edges=[q[i - 1] - p[i], q[i] - q[i - 1], p[i] - q[i]]))
# T'_i is (P'_0,Q'_{i-1},Q'_i)
Tp[i] = s.add_polygon(m * s.polygon(T[i]))
# T'_{g+i} is (P'_i,Q'_i, Q'_{i-1})
Tp[g + i] = s.add_polygon(m * s.polygon(T[g + i]))
for i in range(1, g):
s.change_edge_gluing(T[i], 0, T[i + 1], 2)
s.change_edge_gluing(Tp[i], 2, Tp[i + 1], 0)
for i in range(1, g + 1):
s.change_edge_gluing(T[i], 1, T[g + i], 1)
s.change_edge_gluing(Tp[i], 1, Tp[g + i], 1)
#P 0 Q 0 is paired with P' 0 Q' 0, ...
s.change_edge_gluing(T[1], 2, Tp[g], 2)
s.change_edge_gluing(Tp[1], 0, T[g], 0)
# P1Q1 is paired with P'_g Q_{g-1}
s.change_edge_gluing(T[g + 1], 2, Tp[2 * g], 2)
s.change_edge_gluing(Tp[g + 1], 0, T[2 * g], 0)
# P1Q0 is paired with P_{g-1} Q_{g-1}
s.change_edge_gluing(T[g + 1], 0, T[2 * g - 1], 2)
s.change_edge_gluing(Tp[g + 1], 2, Tp[2 * g - 1], 0)
# PgQg is paired with Q1P2
s.change_edge_gluing(T[2 * g], 2, T[g + 2], 0)
s.change_edge_gluing(Tp[2 * g], 0, Tp[g + 2], 2)
for i in range(2, g - 1):
# PiQi is paired with Q'_i P'_{i+1}
s.change_edge_gluing(T[g + i], 2, Tp[g + i + 1], 2)
s.change_edge_gluing(Tp[g + i], 0, T[g + i + 1], 0)
s.set_immutable()
return TranslationSurface(s)
def cos_minpoly(n, var='x'):
r"""
Return the minimal polynomial of 2 cos pi/n
Done via [KoRoTr2015]_ Algorithm 3.
EXAMPLES::
sage: from flatsurf.geometry.subfield import cos_minpoly
sage: cos_minpoly(1)
x + 2
sage: cos_minpoly(2)
x
sage: cos_minpoly(3)
x - 1
sage: cos_minpoly(4)
x^2 - 2
sage: cos_minpoly(5)
x^2 - x - 1
sage: cos_minpoly(6)
x^2 - 3
sage: cos_minpoly(8)
x^4 - 4*x^2 + 2
sage: cos_minpoly(9)
x^3 - 3*x - 1
sage: cos_minpoly(10)
x^4 - 5*x^2 + 5
sage: cos_minpoly(90)
x^24 - 24*x^22 + 252*x^20 - 1519*x^18 + 5796*x^16 - 14553*x^14 + 24206*x^12 - 26169*x^10 + 17523*x^8 - 6623*x^6 + 1182*x^4 - 72*x^2 + 1
sage: cos_minpoly(148)
x^72 - 72*x^70 + 2483*x^68 - 54604*x^66 + 860081*x^64 ... - 3511656*x^6 + 77691*x^4 - 684*x^2 + 1
"""
n = ZZ(n)
facs = list(n.factor())
if isinstance(var, str):
var = polygen(ZZ, var)
if not facs:
# 2 cos (pi) = -2
return var + 2
if facs[0][0] == 2: # n is even
k = facs[0][1]
facs.pop(0)
else:
k = 0
if not facs:
# 0. n = 2^k
# ([KoRoTr2015] Lemma 12)
return chebyshev_T(2**(k-1), var)
# 1. Compute M_{n0} = M_{p1 ... ps}
# ([KoRoTr2015] Lemma 14 and Lemma 15)
M = cos_minpoly_odd_prime(facs[0][0], var)
for i in range(1, len(facs)):
p = facs[i][0]
M, r = M(chebyshev_T(p, var)).quo_rem(M)
assert r.is_zero()
# 2. Compute M_{2^k p1^{a1} ... ps^{as}}
# ([KoRoTr2015] Lemma 12)
nn = 2**k * prod(p**(a-1) for p,a in facs)
if nn != 1:
M = M(chebyshev_T(nn, var))
return M