def sigma(self, prec=10):
"""
EXAMPLES::
sage: E = EllipticCurve('14a')
sage: F = E.formal_group()
sage: F.sigma(5)
t + 1/2*t^2 + (1/2*c + 1/3)*t^3 + (3/4*c + 3/4)*t^4 + O(t^5)
"""
a1,a2,a3,a4,a6 = self.curve().ainvs()
k = self.curve().base_ring()
fl = self.log(prec)
R = rings.PolynomialRing(k,'c'); c = R.gen()
F = fl.reverse()
S = rings.LaurentSeriesRing(R,'z')
c = S(c)
z = S.gen()
F = F(z + O(z**prec))
wp = self.x()(F)
e2 = 12*c - a1**2 - 4*a2
g = (1/z**2 - wp + e2/12).power_series()
h = g.integral().integral()
sigma_of_z = z.power_series() * h.exp()
T = rings.PowerSeriesRing(R,'t')
fl = fl(T.gen()+O(T.gen()**prec))
sigma_of_t = sigma_of_z(fl)
return sigma_of_t
def sigma(self, prec=10):
r"""
Return the Weierstrass sigma function as a formal power series
solution to the differential equation
.. MATH::
\frac{d^2 \log \sigma}{dz^2} = - \wp(z)
with initial conditions `\sigma(O)=0` and `\sigma'(O)=1`,
expressed in the variable `t=\log_E(z)` of the formal group.
INPUT:
- ``prec`` - integer (default 10)
OUTPUT: a power series with given precision
Other solutions can be obtained by multiplication with
a function of the form `\exp(c z^2)`.
If the curve has good ordinary reduction at a prime `p`
then there is a canonical choice of `c` that produces
the canonical `p`-adic sigma function.
To obtain that, please use ``E.padic_sigma(p)`` instead.
See :meth:`~sage.schemes.elliptic_curves.ell_rational_field.EllipticCurve_rational_field.padic_sigma`
EXAMPLES::
sage: E = EllipticCurve('14a')
sage: F = E.formal_group()
sage: F.sigma(5)
t + 1/2*t^2 + 1/3*t^3 + 3/4*t^4 + O(t^5)
"""
a1, a2, a3, a4, a6 = self.curve().ainvs()
k = self.curve().base_ring()
fl = self.log(prec)
F = fl.reverse()
S = rings.LaurentSeriesRing(k, 'z')
z = S.gen()
F = F(z + O(z**prec))
wp = self.x()(F) + (a1**2 + 4 * a2) / 12
g = (1 / z**2 - wp).power_series()
h = g.integral().integral()
sigma_of_z = z.power_series() * h.exp()
T = rings.PowerSeriesRing(k, 't')
fl = fl(T.gen() + O(T.gen()**prec))
sigma_of_t = sigma_of_z(fl)
return sigma_of_t