Torsion of rational elliptic curves over cubic fields and sporadic points on $X_1(n)$

Abstract

Abstract. We classify the possible torsion structures of rational elliptic curves over cubic fields. Along the way we find a previously unknown torsion structure over a cubic field, Z/21Z, which corresponds to a sporadic point on X1(21) of degree 3, which is the lowest possible degree of a sporadic point on a modular curve X1(n).

Families of elliptic curves with prescribed torsion subgroups over dihedral quartic fields

“…The first two authors [4] proved that X 1 (N ) is bielliptic if and only if N = 13, 16,17,18,20,21,22,24. Some bielliptic involutions of X 1 (N ) are listed in Table 3.…”

“…There are results on torsion groups of elliptic curves over rational number fields [16], quadratic number fields [11][12][13]24,17,18], cubic number fields [10,23,22,19,21], and quartic number fields [9,20,25].…”

On the cyclic torsion of elliptic curves over cubic number fields

“…Najman [27] discovered a sporadic elliptic curve over a cubic field with torsion group isomorphic to Z/21Z. In view of these facts, our ultimate aim is to show that the torsion group E(K) tor of an elliptic curve E over a cubic number field is isomorphic to one of the following: Z/mZ, m = 1 − 16, 18, 20 − 21;…”

“…For the cyclic case, it suffices to show that Z/NZ is not a subgroup of E(K) tor for any elliptic curve E over a cubic number field K when N is among the following list N = 169, 121, 49, 25,27,32; …”

Cartan images and $$\ell $$ ℓ -torsion points of elliptic curves with rational j-invariant

“…Mazur famously classified the possible torsion subgroups E(Q) tors in [25] and the possible -isogenies of an elliptic curve E/Q in [26]. Kamienny, Kenku and Momose generalized Mazur's results on torsion subgroups to quadratic number fields in [22,23], and though no complete characterization for higherdegree number fields is known, there has been recent progress towards characterizing the cubic case [6,17,18,29,30,35], the quartic case [7,14,19,28], and the quintic case [8,12]. In particular, the set of torsion subgroups that arise for infinitely many Q-isomorphism classes of elliptic curves defined over number fields of degree d has been determined for d = 3, 4, 5, 6 [10,20,21].…”