A remark on certain Weierstrass forms (Q2890673)

In this rather computationally-heavy paper the authors describe a simple algorithm to put in Weierstrass form an arbitrary genus 1 curve given by an affine equation either of bidegree (2,2) or of general cubic type. Note that the problem of finding a Weierstrass model for a general genus 1 curve given by an arbitrary affine equation has been treated in [\textit{M. van Hoeij}, in: Proceedings of the 1995 international symposium on symbolic and algebraic computation, ISSAC '95, Montreal, Canada, July 10--12, 1995. Levelt, A. H. M. (ed.), New York, NY: ACM Press. 90--95 (1995; Zbl 0917.14034)].NEWLINENEWLINEThe algorithm is then applied to give examples of elliptic curves over \(\mathbb{Q}\) with a rational 7-torsion point and rank at least 2 (it is known that there are infinitely many distinct isogeny classes of such curves, cf. [\textit{O. Lecacheux}, Acta Arith. 109, No. 2, 131--142 (2003; Zbl 1036.11022)]). The examples in the paper are obtained both by a direct construction and by specialization from a suitable elliptic curve over \(\mathbb{Q}(t)\), curve which is in turn computed using techniques similar to those of Lecacheux's paper, namely finding fibrations of the K3 modular surface \(S_7\).NEWLINENEWLINEUnder some additional (but rather restrictive) conditions it is also shown that the algorithm, when applied to a tempered polynomial \(P(x,y)\) of bidegree (2,2), outputs a tempered Weierstrass form, which is more suitable for some kinds of computations involving the elliptic regulator. Finally, families of tempered polynomials of bidegree (2,2) are given that satisfy the condition.