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Given an integral positive-definite rank $n$ quadratic form $f$, one can use the algorithm in Conway and Sloane (Chapter 15, SPLaG) to efficiently determine if the genus of $f$ contains more than one spinor genus. My question is: given two (integral positive-definite) forms $f,g$ in the same genus, such that the genus contains more than one spinor genus, can one efficiently determine if the forms lie in distinct spinor genera?

I am interested in $n\geq5$.

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Yes: one can write down the explicit local transformations at all primes where they are not both unimodular and evaluate the spinor norms and the local automorphism groups. Magma in fact claims to implement such an algorithm, but I don't have personal experience with it.

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    $\begingroup$ Magma had some problems relating to the prime 2. In a computer run I did from 2004 - 2007. I admit, I did not specifically find spinor norms, I would get the machine to produce representatives for a spinor genus (positive ternary forms). Which is why I am hoping for big things in Sage, I would feel relieved if someone checked my calculation with robust code. $\endgroup$
    – Will Jagy
    Aug 19 at 16:26
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    $\begingroup$ In case of interest, here is the paper arxiv.org/abs/1711.05811 where they use my calculation up to a certain bound; embarrassing if it later turned out I had missed some.... I see, 2018 with Ribet $\endgroup$
    – Will Jagy
    Aug 19 at 16:30
  • $\begingroup$ Closer to the original question: the first indication of a problem in Magma, most likely about high powers of 2, is this email exchange between Manjul Bhargava and Irving Kaplansky, especially 15 June 1999 zakuski.math.utsa.edu/~kap/Kap_Letters_Bhargava_1999.pdf $\endgroup$
    – Will Jagy
    Aug 21 at 0:44
  • $\begingroup$ Thanks to you both for your answers/comments! A follow up question: is anything known about the complexity of computing the local transformations and automorphism groups? And a reference on these computations would be greatly appreciated! $\endgroup$
    – a196884
    Aug 31 at 9:28
  • $\begingroup$ All of this follows from local canonicalization, the algorithm for which is buried in Conway. You shouldn't have a problem for odd primes. As for complexity it's Gram Schmidt and some sorting, and the input size will affect the speed of the multiplications in the usual way. $\endgroup$ Sep 7 at 21:49

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