Is there an implemented algorithm that will take a polynomial $f(x,y)\in\mathbb Z[x,y]$ and a prime $p$, and determine whether the equation $f(x,y)=0$ has a solution over $\mathbb Q_p$?

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My recollection is that Nils Bruin did this is in fair generality in Magma. I do not know how efficient it is.


IsLocallySolvable(X, p) : Sch, RngIntElt -> BoolElt, Pt

    Smooth: BoolElt                     Default: false
    AssumeIrreducible: BoolElt          Default: false
    AssumeNonsingular: BoolElt          Default: false

Given a projective scheme X defined over a number field or over the rationals, test if the scheme is locally solvable at the prime ideal p (for number fields) or prime number p (for rationals) indicated. If the scheme is found to have a local point, then true is returned together with an approximation to a point. Otherwise, false is returned.

    For a description of the algorithms used, see [Bru04]. 
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