## ELLIPTIC CURVES COMING FROM HERON TRIANGLES

### BibTeX

@MISC{Dujella_ellipticcurves,

author = {Andrej Dujella and Juan Carlos Peral},

title = {ELLIPTIC CURVES COMING FROM HERON TRIANGLES},

year = {}

}

### OpenURL

### Abstract

Triangles having rational sides a, b, c and rational area Q are called Heron triangles. Associated to each Heron triangle is the quartic v 2 = u(u − a)(u − b)(u − c). The Heron formula states that Q = √ P (P − a)(P − b)(P − c) where P is the semi-perimeter of the triangle, so the point (u, v) = (P, Q) is a rational point on the quartic. Also the point of in nity is on the quartic. By a standard construction it can be proved that the quartic is equivalent to the elliptic curve y 2 = (x + a b)(x + b c)(x + c a). The point (P, Q) on the quartic transforms to −2abc (x, y) =

### Citations

231 | Advanced Topics in the Arithmetic of Elliptic Curves - Silverman - 1994 |

223 |
Algorithms for modular elliptic curves
- Cremona
- 1992
(Show Context)
Citation Context ...e, are satis ed. We have E(12) : y 2 = x 3 − 51289727495763299303985770723092429x 2 + 421183417712526829656944728081554833692892562346406588197120401049000x.8 A. DUJELLA AND J. C. PERAL Using mwrank =-=[C]-=-, we compute that rank(E(12)(Q)) = 5. Hence, we proved that rank(E(Q(w))) = 5. Moreover, mwrank is able to nd free generators of E(12)(Q) (we increased the default saturation bound from 100 to 350 to ... |

138 |
History of the Theory of Number
- Dickson
- 1952
(Show Context)
Citation Context ... J. C. PERAL Observe that the value of Q is a consequence of the Heron formula. A classical reference for Heron triangles is the second volume of the History of the Theory of Numbers by L. E. Dickson =-=[DI]-=-. 1.2. Elliptic curves associated to Heron triangles. Given a Heron triangle of sides {a, b, c} and area Q, consider the quartic (1) v 2 = u(u − a)(u − b)(u − c). The point (u, v) = ((a+b+c)/2, Q) is ... |

12 | On Mordell-Weil groups of elliptic curves induced by Diophantine triples
- Dujella
(Show Context)
Citation Context ...here exist an in nite set of Heron triangles having two rational medians. Elliptic curves of the shape (2) appear in a natural way in the study of elliptic curves induced by Diophantine m-tuples (see =-=[D]-=- and [ADP]), where the values a, b and c represent three components of a Diophantine triple, instead of the sides of a Heron triangle as in the present context. In [CG] the authors describe connection... |

11 | Construction de courbes elliptiques sur Q de rang ≥ 12 - Mestre - 1982 |

10 | An example of elliptic curve over Q with rank ≥ 20 - Nagao - 1993 |

9 | m-tuples and elliptic curves - Dujella |

7 | An in nite set of Heron triangles with two rational medians
- Buchholz, Ratbun
- 1997
(Show Context)
Citation Context ...angles with a given area has been shown in [R]. This result is also obtained in [GM], by exploiting properties of a family of elliptic curves which generalize the congruent number elliptic curves. In =-=[BR]-=- it is shown that there exist an in nite set of Heron triangles having two rational medians. Elliptic curves of the shape (2) appear in a natural way in the study of elliptic curves induced by Diophan... |

7 | Diophantine triples and construction of high-rank elliptic curves over Q with three non-trivial 2-torsion points - Dujella |

7 |
version 2.4.0
- PARIGP
(Show Context)
Citation Context ...ne these information with the conjectural parity for the rank. Finally, we try to compute the rank and nd generators for the best candidates for large rank. We have implemented this procedure in PARI =-=[P]-=-, using Cremona's program mwrank [C] for the computation of rank and Selmer rank. In the following tables, we present examples of rank 9. We give the corresponding parameters n, w and also the sides o... |

4 |
On rational triangles
- Fine
- 1976
(Show Context)
Citation Context ...so represent a full basis for E(12). Finally, by [GT, Corollary 3.2], we conclude that Q1, . . . , Q5 are free generators of E(Q(w)). 2.6. Rank 9 and 10. In [IKN] Izadi et al., using a result of Fine =-=[F]-=-, found a subfamily of rank ≥ 3 and also several examples of elliptic curves with rank 7 over Q associated to Heron triangles. Here we will give some examples of such curves with rank 9 and 10. Our st... |

4 |
An elliptic K3 surface associated to Heron triangles
- Luijk
(Show Context)
Citation Context ...ed by Bremner in [B] in order to study the existence of sets of N Heron triangles with given perimeter and area, for a given positive integer N. Similar kind of problems have been treated in [KL] and =-=[vL]-=-. The existence of in nitely many Heron triangles with a given area has been shown in [R]. This result is also obtained in [GM], by exploiting properties of a family of elliptic curves which generaliz... |

3 | On the rank of elliptic curves coming from rational Diophantine triples
- Aguirre, Dujella, et al.
(Show Context)
Citation Context ...st an in nite set of Heron triangles having two rational medians. Elliptic curves of the shape (2) appear in a natural way in the study of elliptic curves induced by Diophantine m-tuples (see [D] and =-=[ADP]-=-), where the values a, b and c represent three components of a Diophantine triple, instead of the sides of a Heron triangle as in the present context. In [CG] the authors describe connections between ... |

2 |
triangles via elliptic curves
- Goins, Maddox
(Show Context)
Citation Context ...sitive integer N. Similar kind of problems have been treated in [KL] and [vL]. The existence of in nitely many Heron triangles with a given area has been shown in [R]. This result is also obtained in =-=[GM]-=-, by exploiting properties of a family of elliptic curves which generalize the congruent number elliptic curves. In [BR] it is shown that there exist an in nite set of Heron triangles having two ratio... |

2 |
A remark on the injectivity of the specialization homomorphism
- Gusi¢, Tadi¢
(Show Context)
Citation Context ...qual to 7 over Q. The purpose of this note is to improve their results. In fact, we have found families of elliptic curves induced by Heron triangles having rank ≥ 4 and ≥ 5. Using the algorithm from =-=[GT]-=-, we are able to show that the generic rank of the last family is equal to 5 and to nd generators of its Mordell-Weil group. Furthermore, we have found particular examples of curves whose rank over Q ... |

2 | Rational triangles with equal area
- Rusin
- 1998
(Show Context)
Citation Context ...en perimeter and area, for a given positive integer N. Similar kind of problems have been treated in [KL] and [vL]. The existence of in nitely many Heron triangles with a given area has been shown in =-=[R]-=-. This result is also obtained in [GM], by exploiting properties of a family of elliptic curves which generalize the congruent number elliptic curves. In [BR] it is shown that there exist an in nite s... |

2 |
On Heron triangles, Ann
- Bremner
(Show Context)
Citation Context ...The relation between elliptic curves and Heron triangles appears in the work of many authors in the mathematical literature. Let us mention some of them. The quartics (1) have been used by Bremner in =-=[B]-=- in order to study the existence of sets of N Heron triangles with given perimeter and area, for a given positive integer N. Similar kind of problems have been treated in [KL] and [vL]. The existence ... |

1 | On Heron triangles - Bremner |

1 |
A new family of elliptic curves with positive ranks arising from the Heron triangles, preprint
- Izadi, Khoshnam, et al.
(Show Context)
Citation Context ...les with a given area and nding Diophantine quadruples, and they are led to study the relation of these problems with the elliptic curves over Q having rational torsion group equal to Z/2Z × Z/8Z. In =-=[IKN]-=- the authors show a family of Heron triangles whose associated elliptic curves have generic rank equal to 3 and they also exhibit particular examples of curves with rank equal to 7 over Q. The purpose... |