Possible torsion of elliptic curves over cyclic cubic fields of conductor between 1 and 100

Shankar, G Yuvan; Rakvi,

doi:10.2140/involve.2025.18.897

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Possible torsion of elliptic curves over cyclic cubic fields of conductor between 1 and 100

G Yuvan Shankar and Rakvi

Vol. 18 (2025), No. 5, 897–908

DOI: 10.2140/involve.2025.18.897

Abstract

We provide a list of possible torsion groups that can occur for a cyclic cubic field $K$ where the conductor of $K$ is between $1$ and $100$ . We were not able to rule out or prove the existence of: $ℤ ∕ 13 ℤ$ over cyclic cubic fields of conductor $63$ , $ℤ ∕ 18 ℤ$ over cyclic cubic field of conductor $9$ , or $ℤ ∕ 2 ℤ \times ℤ ∕ 14 ℤ$ over $K \subset ℚ (ζ_{63})$ isomorphic to $ℚ [x] ∕ ⟨ x^{3} - 195 x^{2} + 3267 x - 1 ⟩$ .

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Keywords

elliptic curves, cubic fields, torsion groups

Mathematical Subject Classification

Primary: 11G05, 14H52

Milestones

Received: 5 April 2024

Revised: 28 May 2024

Accepted: 13 June 2024

Published: 20 November 2025

Communicated by Nathan Kaplan

Authors

G Yuvan Shankar
Indian Institute of Technology Guwahati India

Rakvi
Department of Mathematics University of Pennsylvania Philadelphia, PA United States

Vol. 18, No. 5, 2025