CoCalc Public Fileswww / 168 / notes / 2005-09-28 / sage-find-points-on-curve.txt
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
1[email protected]:~/www/168/notes/2005-09-28\$ sage
2------------------------------------------------------------------------
3  SAGE Version 0.7.5, Export Date: 2005-09-29-0607
4    Distributed under the terms of the GNU General Public License (GPL)
5      IPython shell -- for help type <object>?, <object>??, %magic, or help
6      ------------------------------------------------------------------------
7
8       sage: E = EllipticCurve([0,0,1,-1,0])
9       sage: E
10        _4 = Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
11	sage: P = E([0,0])
12	sage: P
13	 _7 = (0, 0)
14	 sage: P+P
15	  _8 = (1, 0)
16	  sage: P+P+P
17	   _9 = (-1, -1)
18	   sage: P+P+P+P
19	   _10 = (2, -3)
20	   sage: P+P+P+P+P
21	   _11 = (1/4, -5/8)
22	   sage: P+P+P+P+P+P
23	   _12 = (6, 14)
24	   sage: 6*P
25	   _13 = (6, 14)
26	   sage: 7*P
27	   _14 = (-5/9, 8/27)
28	   sage: 8*P
29	   _15 = (21/25, -69/125)
30	   sage: 9*P
31	   _16 = (-20/49, -435/343)
32	   sage: 10*P
33	   _17 = (161/16, -2065/64)
34	   sage: 20*P
35	   _18 = (683916417/264517696, -18784454671297/4302115807744)
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