Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Computing the Mazur and Swinnerton-Dyer critical subgroup of elliptic
  curves   Let $E$ be an optimal elliptic curve defined over $\mathbb{Q}$. The critical
subgroup of $E$ is defined by Mazur and Swinnerton-Dyer as the subgroup of
$E(\mathbb{Q})$ generated by traces of branch points under a modular
parametrization of $E$. We prove that for all rank two elliptic curves with
conductor smaller than 1000, the critical subgroup is torsion. First, we define
a family of critical polynomials attached to $E$ and describe two algorithms to
compute such polynomials. We then give a sufficient condition for the critical
subgroup to be torsion in terms of the factorization of critical polynomials.
Finally, a table of critical polynomials is obtained for all elliptic curves of
rank two and conductor smaller than 1000, from which we deduce our result.

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The categories of this paper are: nlin
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The categories of this paper are: atom-ph
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The categories of this paper are: q-bio
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The categories of this paper are: chem-ph
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The categories of this paper are: q-alg
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
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The categories of this paper are: math-ph
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The categories of this paper are: physics
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The categories of this paper are: eess
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The categories of this paper are: hep-lat
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin
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The categories of this paper are: supr-con