I have elliptic curve E : [0,0,0,-3267,45630], point P=[51,-108], as well as point Q=[-57,432] on elliptic curve F : [0,27,0,-4077,51705]. I want to find all (m,n) such that x(mP)=x(nQ). I tried doing it using Pari-gp unfortunately it keeps giving error (I am not sure why). Is there a way I can build this code using Sage.
for(m=1,10,for(n=1,10,XX=ellpow(E,P,m);YY=ellpow(F,Q,n);X=XX[1];Y=YY[1];if(X==Y,print(m "\t" n))))
I guess the second point Q is given by [-57,216]. I use it in what follows.
E=EllipticCurve([0,0,0,-3267,45630])
P=E([51,-108])
Q=E([-57,216])
[(m,n) for m in [1..10] for n in [1..10] if (m*P)[0]==(n*Q)[0]]
The result is:
[(2, 2), (4, 4), (6, 6), (8, 8), (10, 10)]
e.g. 2P=(339 : 6156 : 1) and 2Q=(339 : -6156 : 1).
Be careful: Looking at P[0] doesn't give you the affine x-coordinate. For instance, if you set
E=EllipticCurve([0,1,0,1,0])
P=E([0,0])
Q=E(0)
the code above would erroneously report that P and Q have the same x-coordinate
You could instead use
T=lambda(P,Q): P[0]*Q[2]==P[2]*Q[0]
or test (m*P).xy()[0] == (n*Q).xy()[0] which does look at the affine coordinates.
Thanks @nbruin for the correction, now using the second suggested test we may go as
E=EllipticCurve([0,0,0,-3267,45630])
P=E([51,-108])
F=EllipticCurve([0,27,0,-4077,51705])
Q=F([-57,432])
[(m,n) for m in [1..100] for n in [1..100] if (m*P).xy()[0] == (n*Q).xy()[0]]
the result is simply [].
Asked: 2017-01-24 09:48:54 +0100
Seen: 1,339 times
Last updated: Jan 25 '17
Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.
@Sha - What have you tried?