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- # Lets show that div(f) = [P] - [∞]
- # So any divisor with supp(D) = {P},
- # with an effective size of 1 can be represented
- # by the horizontal line f = y - P.y
- q = 47
- K = GF(q)
- E = EllipticCurve(K, (0, 5))
- C = E.defining_polynomial()
- R.<x, y> = PolynomialRing(K)
- for i in range(100):
- P = E.random_point()
- Px, Py = P[0], P[1]
- # Skip points at infinity
- if P[2] == 0:
- continue
- assert P[2] == 1
- f = y - Py
- I = Ideal([C(x, y, 1), f])
- V = I.variety()
- print(P, V)
- assert len(V) == 1
- assert V[0][x] == Px
- assert V[0][y] == Py
- # Note: to reduce D = [P] - [Q], just note that
- # D = [P] - [Q]
- # = ([P] - [∞]) - ([Q] - [∞])
- # = div(f) - div(g) = div(f/g)
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