show how to calculate intersection_multiplicity of point on elliptic curve: otherwise known as the order of vanishing - ord_P(f)

script/research/ec/intersection-multiplicity.sage

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+# Calculate intersection multiplicity of a point in sage
+q = 47
+K = GF(q)
+E = EllipticCurve(K, (0, 5))
+Q = E(10, 26)
+
+C = E.defining_polynomial()
+
+R.<x, y, z> = PolynomialRing(K)
+f = y - Q[1] * z
+
+P.<x,y,z> = ProjectiveSpace(K, 2)
+X = P.subscheme([C(x, y, z)])
+Y = P.subscheme([f(x, y, z)])
+
+Q = X([Q[0], Q[1]])
+print(Q.intersection_multiplicity(Y))
+