Elliptic curves associated with simplest quartic fields

Lower bounds of the canonical height on quadratic twists of elliptic curves

“…Therefore λ p (Q) = max{0, −v p (α/δ 2 )} log p = 0. At first, we assume that p|c 4 . Then E D has the additive reduction at p. By (4.11), N = v p (∆) = 2, 3 or 4 since ∆ is 6th-power-free.…”

“…In the paper [4,Proposition 8.3], Duquesne gave an explicit lower bound of the canonical heights of raional points on a certain family of elliptic curves. The family consists of quartic twists of the elliptic curve y 2 = x 3 − x.…”

“…Our lower bound is computed by using the decomposition of the canonical height into the local heights and they are computed by the combination of Cohen's algorithm ([2, Algorithm 7.5.7]) and Silverman's algorithm ( [6,Theorem 5.2]). In [4] and [5], by the simplicity of the forms of the Weierstrass equations, the estimates of the nonarchimedean part of the local height were given by ad hoc arguments. However, in our case more systematic argument is required.…”

A Geometric Framework for the Subfield Problem of Generic Polynomials Via Tschirnhausen Transformation

“…with discriminant 4(n 2 + 16) 3 whose Galois group over Q is isomorphic to C 4 except for n = 0, ±3 (cf. for example, [Gra77], [Gra87], [Laz91], [LP95], [Kim04], [HH05], [Duq07], [Lou07], and the references therein). By Lemma 4.2, we see that h n (X) and…”

Generators for the elliptic curve y^2=x^3-nx

“…While in some cases of rank one we determined the generators for E(Q) (e.g., in case k = 1 and p = (2t) 2 + 1 with odd t, E(Q) = (0, 0), (−1, 2t) ), we were not able to do that in rank two cases (e.g., in case k = 1 and p = a 4 + b 4 > 17, the independence of the points (−b 2 , a 2 b) and (−a 2 , ab 2 ) was only found). Duquesne ([5,Theorem 12.3]) remarkably showed that if n = (2k 2 −2k+1)(18k 2 +30k+17) is squarefree with an integer k, then the points G 1 = (−(2k 2 −2k +1), 4(k +1)(2k 2 − 2k + 1)) and G 2 = (9(2k 2 − 2k + 1), 12(3k − 2)(2k 2 − 2k + 1)) can always be in a system of generators for E(Q) (where G 1 and G 2 above correspond to G 2 and G 1 + G 2 in [5], respectively. Note that he also determined the integer points on a quartic form of E assuming rankE(Q) = 2).…”