Symbolic computation of multiples of a point in elliptic curve.

asked 3 years ago

Artatrana
31 ●3

updated 3 years ago

FrédéricC

5426 ●4 ●44 ●121

Suppose we have an elliptic curve over the rational field $E(\mathbb{Q}).$ Let $P = (s,t)\in E(\mathbb{Q})$ be a point with $s,t$ being symbolic variables. Then we see that $n*P = (f_{n1}/g_{n1}, f_{n2}/g_{n2})$ , where $f_{ni}, g_{ni}$ are polynomials in the variables $s,t.$ In SageMath after defining the curve and the symbolic variables $s$ and $t$ if I write $P = E(s,t)$ , it is giving error. I am looking for method by which I can find all these polynomials in the variabls $s,t$ using SageMath.

Comments

Could you please provide your current code so that we could understand the issue ?

tmonteil ( 3 years ago )

Note that if $s,t$ are symbolic variables, then $(s,t)$ is not a pair of rational numbers, so it does not lie in $E(\mathbb{Q})$ . Furthermore, if $s,t$ are just symbolic variables then $t^2\neq s^3+c_4s+c_6$ identically, since they are just independent variables. You'd want to work in a ring or field where you actually have $s,t$ that satisfy the relation required. The function field of $E$ would do.

nbruin ( 3 years ago )

add a comment

def get_poly(E, n): P.<s,t> = PolynomialRing(E.base_ring()) a = E.a_invariants() R = P.quotient_ring(t^2 + (a[0]*s + a[2])*t - (s^3 + a[1]*s^2 + a[3]*s + a[4])).fraction_field() Q = E.change_ring(R)(s,t) * n return Q[0].numerator().lift(), Q[0].denominator().lift(), Q[1].numerator().lift(), Q[1].denominator().lift()

Your Answer

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

Symbolic computation of multiples of a point in elliptic curve.

Comments

1 Answer

Your Answer

Question Tools

Stats

Related questions

Symbolic computation of multiples of a point in elliptic curve. savecancel

Comments

1 Answer

Your Answer

Question Tools

Stats

Related questions

Symbolic computation of multiples of a point in elliptic curve.