I have an elliptic curve $y^2 = x^3 - 5x + 3$. The line $y = -x + 1$ intercepts it in 3 places but if i change the coefficient of x to -3 it only intercepts it in one place. Why is this?
2026-04-02 08:46:23.1775119583
Lines intersecting Elliptic Curves
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Welcome to MSE! An elliptic curve and a line intersect in 0, 1 or 3 points in the projective plane. Note that all points of the curve lie in the affine plane up to the base point (unit element of the group given the points on the curve). This fact is the basis of the group law of the curve (addition of points). The result holds over any alg. closed field.