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How do I convert a 6th degree curve to an elliptic curve?

asked 2020-05-14 04:16:32 -0500

Gamzeee gravatar image

How do I convert a 6th degree curve to an elliptic curve?

exp: How does y^2=16x^6-32x^5+272x^4+4096x^2-8192*x+69632 curve convert to elliptic curve?

Thanks.

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answered 2020-05-14 08:30:56 -0500

dan_fulea gravatar image

The given equation defines (the affine part of) a hyperelliptic curve of genus two:

sage: R.<x> = PolynomialRing(QQ)                                                                 
sage: g = 16*x^6 - 32*x^5 + 272*x^4 + 4096*x^2 - 8192*x + 69632                                  
sage: g.factor()                                                                                 
(16) * (x^2 - 2*x + 17) * (x^4 + 256)
sage: C = HyperellipticCurve(g)                                                                  
sage: C.genus()                                                                                  
2
sage: J = C.jacobian()                                                                           
sage: J.dimension()                                                                              
2
sage: J                                                                                          
Jacobian of Hyperelliptic Curve over Rational Field defined by y^2 = 16*x^6 - 32*x^5 + 272*x^4 + 4096*x^2 - 8192*x + 69632

The genus is a discrete invariant, the given curve can not be (birationally) "converted" into an elliptic curve. There is a map to its Jacobian. For instance, the point $P(2,272)$ is a $\Bbb Q$-rational point of $C$, i.e. $P\in C(\Bbb Q)$, and its image in the Jacobian J is described in sage as...

sage: P = C( [2, 272] )                                                                                  
sage: P                                                                                                  
(2 : 272 : 1)
sage: P in C                                                                                             
True
sage: Q = J(QQ)(P)                                                                                       
sage: Q                                                                                                  
(x - 2, y - 272)
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Asked: 2020-05-14 04:16:32 -0500

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Last updated: May 14