-------------------------------------------------------------
  SAGE Version 0.7.9, Export Date: 2005-10-10-0104
  Distributed under the terms of the GNU General Public Licen
  IPython shell -- for help type <object>?, <object>??, %magi
-------------------------------------------------------------


sage: E = EC(GF(next_prime(10^20)),[-1,0])
sage: E
Elliptic Curve defined by y^2  = x^3 + 100000000000000000038*x over Finite field of size 100000000000000000039
sage: k = E.base_field()
sage: x = k(3)
sage: x^3 - x
24
sage: y = x^3 - x
sage: y.is_square()
False
sage: x = k(5)
sage: y = x^3 - x
sage: y.is_square()
False
sage: x = k(6)
sage: y = x^3 - x
sage: y.is_square()
False
sage: x = k(7)
sage: y = x^3 - x
sage: y.is_square()
False
sage: k(9).is_square()
True
sage: k(1000000000000000000^2).is_square()
True
sage: x = k(10)
sage: y = x^3 - x
sage: y.is_square()
True
sage: y.sqrt()
33675076078257395357
sage: y = y.sqrt()
sage: x
10
sage: y
33675076078257395357
sage: P = E([x,y])
sage: P
(10, 33675076078257395357)
sage: G = P
sage: 100*G
(69671042218536469665, 41102862215489899547)
sage: 10993929032820842384023840928430*G
(94707288393289496854, 4525092221287435086)
sage: 1099392903282084*G
(53500068972413134021, 78697929278876458561)
sage: 10329999392903282084*G
(66209072218228367624, 27781501562744473795)
sage: G+G
(22247474747474747486, 78431820529969068893)
sage: F
Elliptic Curve defined by y^2  = x^3 + 100000000000000000038*x over Rational Field
sage: G
(10, 33675076078257395357)
sage: F.ainvs()
[0, 0, 0, 100000000000000000038, 0]
sage: k.characteristic()
100000000000000000039
sage: x
10
sage: y
33675076078257395357
sage: G
(10, 33675076078257395357)
sage: G + G
(22247474747474747486, 78431820529969068893)