An Algorithm for Determining Torsion Growth of Elliptic Curves

Abstract

We present a fast algorithm that takes as input an elliptic curve defined over $\mathbb{Q}$ and an integer d and returns all the number fields K of degree $d\prime$ dividing d such that $E{(K)}_{\text{tors}}$ contains $E{(F)}_{\text{tors}}$ as a proper subgroup, for all $F⊈K.$ We ran this algorithm on all elliptic curves of conductor less than 400.000 (a total of 2.483.649 curves) and all $d\le 23$ and collected various interesting data. In particular, we find a degree 6 sporadic point on $X_1(4,12),$ which is so far the lowest known degree a sporadic point on $X_1(m,n),$ for $m\ge 2.$

Keywords:

• Elliptic curves
• torsion over number fields

2010 Mathematics Subject Classification:

• 11G05

Acknowledgement

We would like to thank Jeremy Rouse and David Zureick–Brown for sharing some useful data. We also thank John Cremona for providing access to computer facilities on the Number Theory Warwick Grid at University of Warwick, where the main part of the computations were done and for doing a massive check of all our computations, in particular rechecking that all the curves have the torsion growth we claim. We are greatly indebted to the referee for a very careful and helpful report that significantly improved all aspects of this paper.

Declaration of interest

No potential conflict of interest was reported by the author(s).

Notes

1 M. Derickx, A. Etropolski, M. van Hoeij, J. Morrow and D. Zureick-Brown have announced results for d = 3.

2 The second author showed in [33] that the elliptic curve with LMFDB label 162.c3 has torsion subgroup $\mathbb{Z}/21\mathbb{Z}$ defined over the cubic field $\mathbb{Q}{({\zeta }_9)}^{+}=\mathbb{Q}({\zeta }_9+{\zeta }_9^{-1})$ where ζ₉ is a primitive 9-th root of unity.

3 Note that the data for non-CM elliptic curves over $\mathbb{Q}$ in the LMFDB provably includes all p for which the mod-p representation is non-surjective (this has been verified using Zywina’s algorithm [39], see https://www.lmfdb.org/EllipticCurve/Q/Reliability).

4 This can be read off from LMFDB - see the footnote in Lemma 2.9.

5 We have $\mathbb{Q}(E[\mathcal{\ell }])=F_1$ by [37, Lemma 5.17].

6 We use [37, Corollary 5.18] where possible. By [38, Theorem 3.6] we have that $x(Q)=\frac{{\varphi }_{\mathcal{\ell }}(x)}{{\psi }_{\mathcal{\ell }}{(x)}^2},$ for any $P=(x,y)$ such that $Q=\mathcal{\ell }P.$ Using this step is crucial (instead of factoring ${\mathcal{\ell }}^k$-division polynomials) as one uses the polynomial (1(1) $${\varphi }_{\mathcal{\ell }}(x)-x(Q){\psi }_{\mathcal{\ell }}{(x)}^2=g_1(x)·\dots ·g_u(x),$$(1) ) of degree ${\mathcal{\ell }}^2$ (over number fields) instead of factoring (over $\mathbb{Q}$) the primitive ${\mathcal{\ell }}^k$-division polynomial, which is of degree ${\mathcal{\ell }}^{2k-2}({\mathcal{\ell }}^2-1)/2.$

7 Note that the smallest integer B such that for every torsion group T possible over $\mathbb{Q}$ there exists an elliptic curve E with $E{(\mathbb{Q})}_{\text{tors}}=T$ and $N_{\mathbb{Q}}(E)\le B$ is B = 210.

Additional information

Funding

The first author was partially supported by the grant PGC2018–095392–B–I00 (Ministerio de Ciencia, Innovación y Universidades; Agencia Estatal de Investigación; and Fondo Europeo de Desarrollo Regional (FEDER), European Union). The second author gratefully acknowledges support from the QuantiXLie Center of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004) and by the Croatian Science Foundation under the project no. IP-2018-01-1313.