def parameter(self, prec=20):
r"""
Return the Tate parameter `q` such that the curve is isomorphic
over the algebraic closure of `\QQ_p` to the curve
`\QQ_p^{\times}/q^{\ZZ}`.
INPUT:
- ``prec`` - the `p`-adic precision, default is 20.
EXAMPLES::
sage: eq = EllipticCurve('130a1').tate_curve(5)
sage: eq.parameter(prec=5)
3*5^3 + 3*5^4 + 2*5^5 + 2*5^6 + 3*5^7 + O(5^8)
"""
try:
qE = self._q
if qE.absolute_precision() >= prec:
return qE
except AttributeError:
pass
E4 = EisensteinForms(weight=4).basis()[0]
Delta = CuspForms(weight=12).basis()[0]
j = (E4.q_expansion(prec + 3))**3 / Delta.q_expansion(prec + 3)
jinv = (1 / j).power_series()
q_in_terms_of_jinv = jinv.reverse()
R = Qp(self._p, prec=prec)
qE = q_in_terms_of_jinv(R(1 / self._E.j_invariant()))
self._q = qE
return qE
def parameter(self, prec=20):
r"""
Return the Tate parameter `q` such that the curve is isomorphic
over the algebraic closure of `\QQ_p` to the curve
`\QQ_p^{\times}/q^{\ZZ}`.
INPUT:
- ``prec`` - the `p`-adic precision, default is 20.
EXAMPLES::
sage: eq = EllipticCurve('130a1').tate_curve(5)
sage: eq.parameter(prec=5)
3*5^3 + 3*5^4 + 2*5^5 + 2*5^6 + 3*5^7 + O(5^8)
"""
try:
qE = self._q
if qE.absolute_precision() >= prec:
return qE
except AttributeError:
pass
E4 = EisensteinForms(weight=4).basis()[0]
Delta = CuspForms(weight=12).basis()[0]
j = (E4.q_expansion(prec + 3)) ** 3 / Delta.q_expansion(prec + 3)
jinv = (1 / j).power_series()
q_in_terms_of_jinv = jinv.reverse()
R = Qp(self._p, prec=prec)
qE = q_in_terms_of_jinv(R(1 / self._E.j_invariant()))
self._q = qE
return qE
def tate_parameter(E, p, prec = 20, R = None):
if R is None:
R = Qp(p,prec)
jE = E.j_invariant()
E4 = EisensteinForms(weight=4).basis()[0]
Delta = CuspForms(weight=12).basis()[0]
j = (E4.q_expansion(prec+3))**3/Delta.q_expansion(prec+3)
jinv = (1/j).power_series()
q_in_terms_of_jinv = jinv.reversion()
return q_in_terms_of_jinv(R(1/E.j_invariant()))
def getcoords(E, u, prec=20, R=None):
if R is None:
R = u.parent()
u = R(u)
p = R.prime()
jE = E.j_invariant()
# Calculate the Tate parameter
E4 = EisensteinForms(weight=4).basis()[0]
Delta = CuspForms(weight=12).basis()[0]
j = (E4.q_expansion(prec + 7)) ** 3 / Delta.q_expansion(prec + 7)
qE = (1 / j).power_series().reversion()(R(1 / jE))
# Normalize the period by appropriate powers of qE
un = u * qE ** (-(u.valuation() / qE.valuation()).floor())
precn = (prec / qE.valuation()).floor() + 4
# formulas in Silverman II (Advanced Topics in the Arithmetic of Elliptic curves, p. 425)
xx = un / (1 - un) ** 2 + sum(
[
qE ** n * un / (1 - qE ** n * un) ** 2
+ qE ** n / un / (1 - qE ** n / un) ** 2
- 2 * qE ** n / (1 - qE ** n) ** 2
for n in range(1, precn)
]
)
yy = un ** 2 / (1 - un) ** 3 + sum(
[
qE ** (2 * n) * un ** 2 / (1 - qE ** n * un) ** 3
- qE ** n / un / (1 - qE ** n / un) ** 3
+ qE ** n / (1 - qE ** n) ** 2
for n in range(1, precn)
]
)
sk = lambda q, k, pprec: sum([n ** k * q ** n / (1 - q ** n) for n in range(1, pprec + 1)])
n = qE.valuation()
precp = ((prec + 4) / n).floor() + 2
tate_a4 = -5 * sk(qE, 3, precp)
tate_a6 = (tate_a4 - 7 * sk(qE, 5, precp)) / 12
Eq = EllipticCurve([R(1), R(0), R(0), tate_a4, tate_a6])
C2 = (Eq.c4() * E.c6()) / (Eq.c6() * E.c4())
C = R(C2).square_root()
s = (C - R(E.a1())) / R(2)
r = (s * (C - s) - R(E.a2())) / 3
t = (r * (2 * s - C) - R(E.a3())) / 2
return (r + C2 * xx, t + s * C2 * xx + C * C2 * yy)
def getcoords(E, u, prec=20, R=None):
if R is None:
R = u.parent()
u = R(u)
p = R.prime()
jE = E.j_invariant()
# Calculate the Tate parameter
E4 = EisensteinForms(weight=4).basis()[0]
Delta = CuspForms(weight=12).basis()[0]
j = (E4.q_expansion(prec + 7))**3 / Delta.q_expansion(prec + 7)
qE = (1 / j).power_series().reversion()(R(1 / jE))
# Normalize the period by appropriate powers of qE
un = u * qE**(-(u.valuation() / qE.valuation()).floor())
precn = (prec / qE.valuation()).floor() + 4
# formulas in Silverman II (Advanced Topics in the Arithmetic of Elliptic curves, p. 425)
xx = un / (1 - un)**2 + sum([
qE**n * un / (1 - qE**n * un)**2 + qE**n / un /
(1 - qE**n / un)**2 - 2 * qE**n / (1 - qE**n)**2
for n in range(1, precn)
])
yy = un**2 / (1 - un)**3 + sum([
qE**(2 * n) * un**2 / (1 - qE**n * un)**3 - qE**n / un /
(1 - qE**n / un)**3 + qE**n / (1 - qE**n)**2 for n in range(1, precn)
])
sk = lambda q, k, pprec: sum(
[n**k * q**n / (1 - q**n) for n in range(1, pprec + 1)])
n = qE.valuation()
precp = ((prec + 4) / n).floor() + 2
tate_a4 = -5 * sk(qE, 3, precp)
tate_a6 = (tate_a4 - 7 * sk(qE, 5, precp)) / 12
Eq = EllipticCurve([R(1), R(0), R(0), tate_a4, tate_a6])
C2 = (Eq.c4() * E.c6()) / (Eq.c6() * E.c4())
C = R(C2).square_root()
s = (C - R(E.a1())) / R(2)
r = (s * (C - s) - R(E.a2())) / 3
t = (r * (2 * s - C) - R(E.a3())) / 2
return (r + C2 * xx, t + s * C2 * xx + C * C2 * yy)
