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I tried to solve the integer points of $y(y+2)=x^3+(x+3)(x+5)$ by using Sage's command E.integral_points(). Its output was $[(0 : 3 : 1)]$. I tried that $(x,y)=(0,3)$ satisfies my equation. But what does the third number $1$ means in Sage's notation?

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2 Answers 2

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It means that it homogenises it and treats it as a curve in projective space.

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Your question is not a mathematics question but a Sage question!

The output for E.integral_points() is a list of points, in your example a list with one element. If you extract that point like this:

sage: E = EllipticCurve([0,1,2,8,15])
sage: E
Elliptic Curve defined by y^2 + 2*y = x^3 + x^2 + 8*x + 15 over Rational Field
sage: E.integral_points()
[(0 : 3 : 1)]
sage: P = E.integral_points()[0]

then you can view it in a different way:

sage: P.xy()
(0, 3)
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