Infinite families of elliptic curves over Dihedral quartic number fields

Abstract

We find infinite families of elliptic curves over quartic number fields with torsion group Z/NZ with N = 20, 24. We prove that for each elliptic curve E t in the constructed families, the Galois group Gal(L/Q) is isomorphic to the Dihedral group D 4 of order 8 for the Galois closure L of K over Q, where K is the defining field of (E t , Q t ) and Q t is a point of E t of order N. We also notice that the plane model for the modular curve X 1 (24) found in Jeon et al. (2011) [1] is in the optimal form, which was the missing case in Sutherland's work (Sutherland, 2012 [12]).

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

“…Regarding notation, see (3.3, 3.4) Proof Let H be an index-2 subgroup of G, −1 not containing − 1, and suppose 7,11,19,43, 67, 163},…”

“…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].…”

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

“…In fact, all the cyclic torsion groups of elliptic curves which occur over quartic number fields (but not over quadratic number fields) are Z/N Z with N = 17, 20, 21, 22, 24. The cases of N = 20 and 24 are treated in [8], and the current work completes the construction of infinite families of elliptic curves over dihedral quartic number fields with cyclic torsion groups (which do not occur over quadratic number fields).…”

“…Recently there have been some developments on the construction of the infinite families of elliptic curves over number fields with the prescribed torsion groups [6][7][8]14,24,26]. 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].…”

“…In [8] we use the 2-divisibility method for N = 20 and 24, and we use the fact that X 1 (N/2) has infinitely many Q-rational points. However, this 2-divisibility method cannot be applied to the rest of the cases such as N = 17, 21, 22; for the cases N = 17 and 21, because N is odd, and for the case N = 22, because X 1 (11) has no non-cuspidal Q-rational points.…”