Congruent Number Elliptic Curves with Rank at Least Three

2010 ◽  
Vol 53 (4) ◽  
pp. 661-666 ◽  
Author(s):  
Jennifer A. Johnstone ◽  
Blair K. Spearman
Keyword(s):  
Elliptic Curves ◽  
Infinite Family ◽  
Congruent Number

AbstractWe give an infinite family of congruent number elliptic curves each with rank at least three.

Elliptic curves with good reduction everywhere over cubic fields

2015 ◽  
Vol 11 (04) ◽  
pp. 1149-1164 ◽  
Author(s):  
Nao Takeshi
Keyword(s):  
Elliptic Curves ◽  
Infinite Family ◽  
Good Reduction ◽  
Cubic Fields

We give a criterion for cubic fields over which there exist no elliptic curves with good reduction everywhere, and we construct a certain infinite family of cubic fields over which there exist elliptic curves with good reduction everywhere.

scholarly journals Heron quadrilaterals with sides in arithmetic or geometric progression

1999 ◽  
Vol 59 (2) ◽  
pp. 263-269 ◽  
Author(s):  
R.H. Buchholz ◽  
J.A. MacDougall
Keyword(s):  
Elliptic Curves ◽  
Arithmetic Progression ◽  
Infinite Family ◽  
Rational Points ◽  
Arithmetic Progressions ◽  
Geometric Progression ◽  
Complete Characterisation

We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterisation is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose sides form either a geometric or an arithmetic progression. The solution of both quadrilateral cases involves searching for rational points on certain elliptic curves.

scholarly journals Right triangles with algebraic sides and elliptic curves over number fields

2009 ◽  
Vol 59 (3) ◽  
Author(s):  
E. Girondo ◽  
G. González-Diez ◽  
E. González-Jiménez ◽  
R. Steuding ◽  
J. Steuding
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Infinite Order ◽  
Number Fields ◽  
Explicit Construction ◽  
Congruent Number ◽  
Number Problem ◽  
Weil Group ◽  
Congruent Number Problem

AbstractGiven any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P λ of infinite order in the Mordell-Weil group of the elliptic curve Y 2 = X 3 − n 2 X over ℚ(λ).

scholarly journals On high rank $\pi/3$ and $2\pi/3$-congruent number elliptic curves

2014 ◽  
Vol 44 (6) ◽  
pp. 1867-1880 ◽  
Author(s):  
A.S. Janfada ◽  
S. Salami ◽  
A. Dujella ◽  
J.C. Peral
Keyword(s):  
Elliptic Curves ◽  
High Rank ◽  
Congruent Number

scholarly journals High rank elliptic curves induced by rational Diophantine triples

2020 ◽  
Vol 55 (2) ◽  
pp. 237-252
Author(s):  
Andrej Dujella ◽  
◽  
Juan Carlos Peral ◽  
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Infinite Family ◽  
New Method ◽  
High Rank ◽  
Perfect Squares ◽  
Diophantine Triple ◽  
Diophantine Triples

A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over ℚ with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank ≥ 7, which are both the current records for that kind of curves.

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