def __scale_by_periods_only__(self):
r"""
If we fail to scale with ``_find_scaling_L_ratio``, we drop here
to try and find the scaling by the quotient of the
periods to the `X_0`-optimal curve. The resulting ``_scaling``
is not guaranteed to be correct, but could well be.
EXAMPLES::
sage: E = EllipticCurve('19a1')
sage: m = E.modular_symbol(sign=+1)
sage: m.__scale_by_periods_only__()
Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by -1, 2 or -2.
sage: m._scaling
1
sage: E = EllipticCurve('19a2')
sage: m = E.modular_symbol(sign=+1)
sage: m._scaling
3/2
sage: m.__scale_by_periods_only__()
Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by -1, 2 or -2.
sage: m._scaling
3
"""
# we only do this inside the cremona-tables.
try:
crla = parse_cremona_label(self._E.label())
except RuntimeError: # raised when curve is outside of the table
print(
"Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by a rational number."
)
self._scaling = 1
else:
print(
"Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by -1, 2 or -2."
)
cr0 = Integer(crla[0]).str() + crla[1] + '1'
E0 = EllipticCurve(cr0)
if self._sign == 1:
q = E0.period_lattice().basis()[0] / self._E.period_lattice(
).basis()[0]
else:
q = E0.period_lattice().basis()[1].imag(
) / self._E.period_lattice().basis()[1].imag()
if E0.real_components() == 1:
q *= 2
if self._E.real_components() == 1:
q /= 2
q = ZZ(int(round(q * 200))) / 200
verbose('scale modular symbols by %s' % q)
self._scaling = q
def _find_scaling_period(self):
r"""
Uses the integral period map of the modular symbol implementation in sage
in order to determine the scaling. The resulting modular symbol is correct
only for the `X_0`-optimal curve, at least up to a possible factor +-1 or +-2.
EXAMPLES::
sage: E = EllipticCurve('11a1')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(1/5, 1)
sage: E = EllipticCurve('11a2')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(1, 5)
sage: E = EllipticCurve('121b2')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(0, 11/2, 0, 11/2, 11/2, 0, 0, -3, 2, 1/2, -1, 3/2)
"""
P = self._modsym.integral_period_mapping()
self._e = P.matrix().transpose().row(0)
self._e /= 2
E = self._E
try:
crla = parse_cremona_label(E.label())
except RuntimeError: # raised when curve is outside of the table
self._scaling = 1
else:
cr0 = Integer(crla[0]).str() + crla[1] + '1'
E0 = EllipticCurve(cr0)
if self._sign == 1:
q = E0.period_lattice().basis()[0] / E.period_lattice().basis(
)[0]
else:
q = E0.period_lattice().basis()[1].imag() / E.period_lattice(
).basis()[1].imag()
if E0.real_components() == 1:
q *= 2
if E.real_components() == 1:
q /= 2
q = QQ(int(round(q * 200))) / 200
verbose('scale modular symbols by %s' % q)
self._scaling = q
self._e *= self._scaling
def _find_scaling_period(self):
r"""
Uses the integral period map of the modular symbol implementation in sage
in order to determine the scaling. The resulting modular symbol is correct
only for the `X_0`-optimal curve, at least up to a possible factor +-1 or +-2.
EXAMPLES::
sage: E = EllipticCurve('11a1')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(1/5, 1)
sage: E = EllipticCurve('11a2')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(1, 5)
sage: E = EllipticCurve('121b2')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(0, 11/2, 0, 11/2, 11/2, 0, 0, -3, 2, 1/2, -1, 3/2)
"""
P = self._modsym.integral_period_mapping()
self._e = P.matrix().transpose().row(0)
self._e /= 2
E = self._E
try :
crla = parse_cremona_label(E.label())
except RuntimeError: # raised when curve is outside of the table
self._scaling = 1
else :
cr0 = Integer(crla[0]).str() + crla[1] + '1'
E0 = EllipticCurve(cr0)
if self._sign == 1:
q = E0.period_lattice().basis()[0]/E.period_lattice().basis()[0]
else:
q = E0.period_lattice().basis()[1].imag()/E.period_lattice().basis()[1].imag()
if E0.real_components() == 1:
q *= 2
if E.real_components() == 1:
q /= 2
q = QQ(int(round(q*200)))/200
verbose('scale modular symbols by %s'%q)
self._scaling = q
self._e *= self._scaling
def __scale_by_periods_only__(self):
r"""
If we fail to scale with ``_find_scaling_L_ratio``, we drop here
to try and find the scaling by the quotient of the
periods to the `X_0`-optimal curve. The resulting ``_scaling``
is not guaranteed to be correct, but could well be.
EXAMPLES::
sage: E = EllipticCurve('19a1')
sage: m = E.modular_symbol(sign=+1)
sage: m.__scale_by_periods_only__()
Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by -1, 2 or -2.
sage: m._scaling
1
sage: E = EllipticCurve('19a2')
sage: m = E.modular_symbol(sign=+1)
sage: m._scaling
3/2
sage: m.__scale_by_periods_only__()
Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by -1, 2 or -2.
sage: m._scaling
3
"""
# we only do this inside the cremona-tables.
try :
crla = parse_cremona_label(self._E.label())
except RuntimeError: # raised when curve is outside of the table
print("Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by a rational number.")
self._scaling = 1
else :
print("Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by -1, 2 or -2.")
cr0 = Integer(crla[0]).str() + crla[1] + '1'
E0 = EllipticCurve(cr0)
if self._sign == 1:
q = E0.period_lattice().basis()[0]/self._E.period_lattice().basis()[0]
else:
q = E0.period_lattice().basis()[1].imag()/self._E.period_lattice().basis()[1].imag()
if E0.real_components() == 1:
q *= 2
if self._E.real_components() == 1:
q /= 2
q = ZZ(int(round(q*200)))/200
verbose('scale modular symbols by %s'%q)
self._scaling = q
def _find_scaling_period(self):
r"""
Uses the integral period map of the modular symbol implementation in sage
in order to determine the scaling. The resulting modular symbol is correct
only for the `X_0`-optimal curve, at least up to a possible factor +- a
power of 2.
EXAMPLES::
sage: E = EllipticCurve('11a1')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(1/5, 1/2)
sage: E = EllipticCurve('11a2')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(1, 5/2)
sage: E = EllipticCurve('121b2')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(0, 0, 0, 11/2, 11/2, 11/2, 11/2, -3, 3/2, 1/2, -1, 2)
TESTS::
sage: E = EllipticCurve('19a1')
sage: m = E.modular_symbol(sign=+1, implementation='sage', normalize='none')
sage: m._find_scaling_period()
sage: m._scaling
1
sage: E = EllipticCurve('19a2')
sage: m = E.modular_symbol(sign=+1, implementation='sage', normalize='none')
sage: m._scaling
1
sage: m._find_scaling_period()
sage: m._scaling
3
"""
P = self._modsym.integral_period_mapping()
self._e = P.matrix().transpose().row(0)
self._e /= 2
E = self._E
try:
crla = parse_cremona_label(E.label())
except RuntimeError: # raised when curve is outside of the table
print(
"Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by a rational number."
)
self._scaling = 1
else:
cr0 = Integer(crla[0]).str() + crla[1] + '1'
E0 = EllipticCurve(cr0)
if self._sign == 1:
q = E0.period_lattice().basis()[0] / E.period_lattice().basis(
)[0]
else:
q = E0.period_lattice().basis()[1].imag() / E.period_lattice(
).basis()[1].imag()
if E0.real_components() == 1:
q *= 2
if E.real_components() == 1:
q /= 2
q = QQ((q * 200).round()) / 200
verbose('scale modular symbols by %s' % q)
self._scaling = q
c = self(0) # required, to change base point from oo to 0
if c < 0:
c *= -1
self._scaling *= -1
self._at_zero = c
self._e *= self._scaling
def _find_scaling_period(self):
r"""
Uses the integral period map of the modular symbol implementation in sage
in order to determine the scaling. The resulting modular symbol is correct
only for the `X_0`-optimal curve, at least up to a possible factor +- a
power of 2.
EXAMPLES::
sage: E = EllipticCurve('11a1')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(1/5, 1)
sage: E = EllipticCurve('11a2')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(1, 5)
sage: E = EllipticCurve('121b2')
sage: m = sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbolSage(E,+1,normalize='period')
sage: m._e
(0, 11/2, 0, 11/2, 11/2, 0, 0, -3, 2, 1/2, -1, 3/2)
TESTS::
sage: E = EllipticCurve('19a1')
sage: m = E.modular_symbol(sign=+1, implementation='sage', normalize='none')
sage: m._find_scaling_period()
sage: m._scaling
1
sage: E = EllipticCurve('19a2')
sage: m = E.modular_symbol(sign=+1, implementation='sage', normalize='none')
sage: m._scaling
1
sage: m._find_scaling_period()
sage: m._scaling
3
"""
P = self._modsym.integral_period_mapping()
self._e = P.matrix().transpose().row(0)
self._e /= 2
E = self._E
try :
crla = parse_cremona_label(E.label())
except RuntimeError: # raised when curve is outside of the table
print("Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by a rational number.")
self._scaling = 1
else :
cr0 = Integer(crla[0]).str() + crla[1] + '1'
E0 = EllipticCurve(cr0)
if self._sign == 1:
q = E0.period_lattice().basis()[0]/E.period_lattice().basis()[0]
else:
q = E0.period_lattice().basis()[1].imag()/E.period_lattice().basis()[1].imag()
if E0.real_components() == 1:
q *= 2
if E.real_components() == 1:
q /= 2
q = QQ(int(round(q*200)))/200
verbose('scale modular symbols by %s'%q)
self._scaling = q
c = self(0) # required, to change base point from oo to 0
if c<0:
c *= -1
self._scaling *= -1
self._at_zero = c
self._e *= self._scaling