We briefly describe how to get the power of logarithm in the asymptotic for the number of vanishings in the family of even quadratic twists of a given elliptic curve. There are four different possibilities, largely dependent on the rational 2-torsion structure of the curve we twist. 1.

Let n be an integer such that the modular curve X0(n) is hyperelliptic of genus ≥ 2 and such that the Jacobian of X0(n) has rank 0 over Q. We determine all points of X0(n) defined over quadratic fields, and we give a moduli interpretation of these points. As a consequence, we show that up to Q-isomorphism, all but finitely many elliptic curves with n-isogenies over quadratic fields are in fact ...

We give explicit examples of infinite families of elliptic curves E over Q with (nonconstant) quadratic twists over Q(t) of rank at least 2 and 3. We recover some results announced by Mestre, as well as some additional families. Suppose D is a squarefree integer and let rE(D) denote the rank of the quadratic twist of E by D. We apply results of Stewart and Top to our examples to obtain results ...

In this paper we investigate Hesse’s elliptic curves Hμ : U + V 3 + W 3 = 3μUVW,μ ∈ Q − {1}, and construct their twists, Hμ,t over quadratic fields, and H̃(μ, t), μ, t ∈ Q over the Galois closures of cubic fields. We also show that Hμ is a twist of H̃(μ, t) over the related cubic field when the quadratic field is contained in the Galois closure of the cubic field. We utilize a cubic polynomial, R...

By considering families of periodic pseudo-orbits on the cylindrical phase space of an area-preserving twist map, we construct curves of rational rotation number that are critical points of a non-negative error norm (the mean-square action gradient or "quadratic flux"). Two alternative angle-parametrizations of the curves are presented and numerical evidence is given that the curves possess the...

The aim of the present study is to determine the feasibility of making a rectangular twist waveguide used to rotate electromagnetic waves. For this purpose, the process of fabricating an aluminum rectangular twist waveguide was simulated by making use of finite element method and Deform software. The optimum length and angle of the twist die for manufacturing a twist waveguide with inner cross ...

Let E be an elliptic curve over ij. A celebrated theorem of Mordell asserts that E(ij) , the (abelian) group of rational points of E, is finitely generated. By the rank of E we mean the rank of E(ij). Thus the rank of E is positive if and only if E possesses an infinity of rational points. Relatively few general qualitative assertions can be made about the rank as E varies. How large can the ra...

Electro-optic measurements were performed on a lamellar nematic phase in which the mesogenic moieties lie in lamellae that are separated by partially perfluorinated side groups. The twist elastic constant K22, viscosity gamma(1), and the quadratic and quartic anchoring strength coefficients are reported. K22 and gamma(1) are found to be considerably smaller than that of typical three-dimensiona...

Given two elliptic curves defined over a number field K, not both with j-invariant zero, we show that there are infinitely many D ∈ K × with pairwise distinct image in K × /K × 2 , such that the quadratic twist of both curves by D have positive Mordell-Weil rank. The proof depends on relating the values of pairs of cubic polynomials to rational points on another elliptic curve, and on a fiber p...