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October 3, 20250 citationsOpen Access

Divisibility of the coefficients of modular polynomials

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FBFlorian Breuer

Key Points

  • The coefficients of modular polynomials are highly divisible by small primes, especially at certain parameters.
  • For algebraic numbers like J=0 and J=1728, unique divisibility properties are revealed.
  • Cyclic isogenies of degree N create vanishing pairs in modular polynomials, impacting coefficient divisibility.
  • Understanding these divisibility patterns can inform studies in number theory and elliptic curves.

Abstract

Let N>1 and let ΦN (X, Y), Y be the modular polynomial which vanishes precisely at pairs of j-invariants of elliptic curves linked by a cyclic isogeny of degree N. In this note we study the divisibility of the coefficients ΦN (X+J, Y+J) for certain algebraic numbers J, in particular J=0 and J=1728. It turns out that these coefficients are highly divisible by small primes at which J is supersingular.

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Cite This Study

Florian Breuer (2025) studied this question.

synapsesocial.com/papers/68e02f3cf0e39f13e7fa27a4https://doi.org/10.48550/arxiv.2509.06423
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