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How does one find solutions to a polynomial over a finite field?

asked 2013-02-15 13:05:04 -0500

Zaubertrank gravatar image

updated 2015-01-14 07:56:28 -0500

FrédéricC gravatar image

I'm trying to find the solutions to the polynomial $y^2=x^3+1$ over $\mathbb{F}_5$. I have constructed the correct polynomial ring, but I don't know what the analogous function to .roots() is for the two variable case. Thanks

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answered 2013-02-16 00:29:17 -0500

Francis Clarke gravatar image

Alternatively

sage: R.<x,y> = GF(5)[]
sage: C = Curve(y^2 - x^3 - 1)
sage: C.point_set().points()
[(0, 1), (0, 4), (2, 2), (2, 3), (4, 0)]
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answered 2013-02-15 23:16:04 -0500

achrzesz gravatar image
sage: E=EllipticCurve(GF(5),[0,1]);E
Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 5
sage: pp=E.points()
sage: [p.xy() for p in pp[1:]]
[(0, 1), (0, 4), (2, 2), (2, 3), (4, 0)]
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Asked: 2013-02-15 13:05:04 -0500

Seen: 256 times

Last updated: Feb 16 '13