# How to check two curves on birational equivalence?

I have two curves, for example hyperelliptic:

y^2 = x^6 + 14*x^4 + 5*x^3 + 14*x^2 + 18
y^2 = x^6 + 14*x^4 + 5*x^3 + 14*x^2 + 5*x + 1


How to check them on birational equivalence (is able one curve be birationally transformed to another?) via Sage?

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Two projective algebraic curves are birationally equivalent if and only if they are isomorphic (Ch 1, Sec 6 of Hartshorne). You can input both of the above curves into Sage and compute various isomorphism class invariants to see that the curves are not isomorphic over C, hence not birationally equivalent. This paper http://www.warwick.ac.uk/~masjap/amsigusa.pdf has some basic facts about Igusa invariants in it.

sage: R.<x> = QQ[]
sage: C = HyperellipticCurve(x^6 + 14*x^4 + 5*x^3 + 14*x^2 + 18)
sage: D = HyperellipticCurve(x^6 + 14*x^4 + 5*x^3 + 14*x^2 + 5*x + 1)
sage: C.igusa_clebsch_invariants()
(-116896, 396482560, -4236950421504, -19899663217714003968)
sage: D.igusa_clebsch_invariants()
(-51616, 79673344, -1252989108224, -21978176867532800)

more

Thanks! But, as I can see, this method can help only if they aren't equivalent. Is it possible to check two curves on birational equivalence in general case?

( 2011-12-22 07:54:20 +0200 )edit

According to your link we should check absolute igusa invariants! We should use .absolute_igusa_invariants_wamelen() or .absolute_igusa_invariants_kohel() instead of .igusa_clebsch_invariants(). And it's true. I can give you two birational equivalent curves but they have different igusa_clebsch_invariants

( 2012-12-24 09:16:21 +0200 )edit