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Every hyperelliptic curve is a degree-2 cover of the projective line. The Weierstrass points of the curve are the ramification points of this cover, of which there are $2g+2$, where $g$ is the genus of the curve.

Over a field of characteristic not 2, a hyperelliptic curve of genus $g$ always has a model of the form $y^2=f(x)$, where $\deg f = 2g+2$, in which case the Weierstrass points are simply the roots of $f(x)$, of which any number between $0$ and $2g+2$ may be rational (defined over the field of definition of the curve).

For curves with a rational Weierstrass point, by moving this point to infinity one can put the curve in the form $y^2=f(x)$ with $\deg f =2g+1$, but otherwise not.

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  • Review status: reviewed
  • Last edited by John Cremona on 2018-05-24 17:10:10
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