A short remark on the script l <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambrid

Abstract In a 2008 paper Ellenberg suggested a strategy to improve the known upper bounds for the script l <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>ℓ</mml:mi> </mml:math> $\ell$ -torsion part of class groups of number fields of fixed degree d . Motivated by this he proposed a question about the number of primitive elements of small height in a number field. Here we answer Ellenberg’s question. We also improve Heath-Brown’s bound for the script l <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>ℓ</mml:mi> </mml:math> $\ell$ -torsion part of class groups of purely cubic number fields, and we generalise our improvement to pure fields of arbitrary odd degree d .