#### Vol. 15, No. 7, 2021

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### Maarten Derickx, Anastassia Etropolski, Mark van Hoeij, Jackson S. Morrow and David Zureick-Brown

Vol. 15 (2021), No. 7, 1837–1864
DOI: 10.2140/ant.2021.15.1837
##### Abstract

Let $K$ be a number field, and let $E∕K$ be an elliptic curve over $K$. The Mordell–Weil theorem asserts that the $K$-rational points $E\left(K\right)$ of $E$ form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of $E\left(K\right)$ for $K$ a cubic number field.

To do so, we determine the cubic points on the modular curves ${X}_{1}\left(N\right)$ for

$N=21,22,24,25,26,28,30,32,33,35,36,39,45,65,121.$

As part of our analysis, we determine the complete lists of $N$ for which ${J}_{0}\left(N\right)$, ${J}_{1}\left(N\right)$, and ${J}_{1}\left(2,2N\right)$ have rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on ${J}_{1}\left(N\right)\left(ℚ\right)$ is generated by $Gal$($\stackrel{̄}{ℚ}∕ℚ$)-orbits of cusps of ${X}_{1}{\left(N\right)}_{\stackrel{̄}{ℚ}}$ for $N\le 55$, $N\ne 54$.

##### Keywords
modular curves, elliptic curves, finitely many cubic points
##### Mathematical Subject Classification
Primary: 11G18
Secondary: 11G05, 11Y50, 14H45
##### Supplementary material

Code for sporadic cubic torsion computations