Ask Your Question
1

is there a bug in height pairing matrix?

asked 2024-12-06 14:53:57 +0100

Sud S gravatar image

updated 2024-12-07 11:30:41 +0100

FrédéricC gravatar image

Hello all,

I tried the following on sagecell online

K.<z2> = NumberField(x^2+1)

E = EllipticCurve(K, [3,105/16])

P = E.point((7,-77/4))

E.height_pairing_matrix([P,P])

and I get the out put

[2.13279604624597 2.82594322680591]

[2.82594322680591 2.13279604624597]

Shouldn't all the coefficients in output matrix be the same given that P and P are linearly dependent?

Thank you in advance!

edit retag flag offensive close merge delete

1 Answer

Sort by » oldest newest most voted
0

answered 2024-12-06 19:53:41 +0100

rburing gravatar image

The construction goes like this (obtained by entering E.height_pairing_matrix?? into a SageMath session):

for j in range(r):
    mat[j, j] = points[j].height(precision=precision, normalised=normalised)
for j in range(r):
    for k in range(j + 1, r):
        mat[j, k] = ((points[j] + points[k]).height(precision=precision, normalised=normalised) - mat[j, j] - mat[k, k]) / 2
        mat[k, j] = mat[j, k]

So the matrix contains P.height() and ((2*P).height() - 2*P.height())/2.

edit flag offensive delete link more

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

1 follower

Stats

Asked: 2024-12-06 14:53:57 +0100

Seen: 42 times

Last updated: Dec 06