Coverage for local/lib/python2.7/site-packages/sage/schemes/elliptic_curves/ell_padic_field.py : 94%

""" 

Elliptic curves over padic fields 

""" 

from __future__ import absolute_import 

#***************************************************************************** 

# Copyright (C) 2007 Robert Bradshaw <robertwb@math.washington.edu> 

# William Stein <wstein@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

import sage.rings.ring as ring 

from .ell_field import EllipticCurve_field 

from . import ell_point 

from sage.rings.all import PolynomialRing 

# Elliptic curves are very different than genus > 1 hyperelliptic curves, 

# there is an "is a" relationship here, and common implementation with regard 

# Coleman integration. 

from sage.schemes.hyperelliptic_curves.hyperelliptic_padic_field import HyperellipticCurve_padic_field 

import sage.databases.cremona 

class EllipticCurve_padic_field(EllipticCurve_field, HyperellipticCurve_padic_field): 

""" 

Elliptic curve over a padic field. 

EXAMPLES:: 

sage: Qp=pAdicField(17) 

sage: E=EllipticCurve(Qp,[2,3]); E 

Elliptic Curve defined by y^2 = x^3 + (2+O(17^20))*x + (3+O(17^20)) over 17-adic Field with capped relative precision 20 

sage: E == loads(dumps(E)) 

True 

""" 

_point = ell_point.EllipticCurvePoint_field 

_genus = 1 

def frobenius(self, P=None): 

""" 

Returns the Frobenius as a function on the group of points of 

this elliptic curve. 

EXAMPLES:: 

sage: Qp=pAdicField(13) 

sage: E=EllipticCurve(Qp,[1,1]) 

sage: type(E.frobenius()) 

<... 'function'> 

sage: point=E(0,1) 

sage: E.frobenius(point) 

(0 : 1 + O(13^20) : 1 + O(13^20)) 

""" 

try: 

_frob = self._frob 

except AttributeError: 

K = self.base_field() 

p = K.prime() 

x = PolynomialRing(K, 'x').gen(0) 

a1, a2, a3, a4, a6 = self.a_invariants() 

if a1 != 0 or a2 != 0: 

raise NotImplementedError("Curve must be in weierstrass normal form.") 

f = x*x*x + a2*x*x + a4*x + a6 

h = (f(x**p) - f**p) 

# internal function: I don't know how to doctest it... 

def _frob(P): 

x0 = P[0] 

y0 = P[1] 

uN = (1 + h(x0)/y0**(2*p)).sqrt() 

yres=y0**p * uN 

xres=x0**p 

if (yres-y0).valuation() == 0: 

yres=-yres 

return self.point([xres,yres, K(1)]) 

self._frob = _frob 

if P is None: 

return _frob 

else: 

return _frob(P)