## ELLIPTIC CURVES AND TRIANGLES WITH THREE RATIONAL MEDIANS

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@MISC{Dujella_ellipticcurves,

author = {Andrej Dujella and Juan and Carlos Peral},

title = {ELLIPTIC CURVES AND TRIANGLES WITH THREE RATIONAL MEDIANS},

year = {}

}

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### Abstract

Abstract. In his paper Triangles with three rational medians, about the characterization of all rational-sided triangles with three rational medians, Buchholz proves that each such triangle corresponds to a point on a oneparameter family of elliptic curves whose rank is at least 2. We prove that in fact the exact rank of the family in Buchholz paper is 3. We also exhibit a subfamily whose rank is at least 4 and we prove the existence of in nitely many curves of rank 5 over Q parametrized by an elliptic curve of positive rank. Finally, we show particular examples of curves within those families having rank 9 and 10 over Q. 1.

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259 |
Algorithms for modular elliptic curves
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- 1997
(Show Context)
Citation Context ...m) 2 , 72(1 + m) 2 (9 + 8m + m 2 )). For the value m = 5 the curve has rank 3 and the three points became P1(5) = (1440, 60480) P2(5) = (−128, 34048) P3(5) = (2592, 191808). A calculation with mwrank =-=[5]-=- shows that these three points are independent and since the specialization map is a homomorphism, we get that the rank over Q(m) is ≥ 3. 2.2. Rank over Q(m) is equal to 3. We will now show that the r... |

152 |
History of the theory of numbers
- Dickson
- 1934
(Show Context)
Citation Context ...ntroduction The existence and parametrization of rational-sided triangles with additional conditions has a long history, see the second volume of the History of the Theory of Numbers by L. E. Dickson =-=[7]-=- for older results. There exist also a more recent and extensive mathematical literature on these topics. See for example [2], [8] and [9] and the references given there. The existence of in nitely ma... |

12 | On Mordell-Weil groups of elliptic curves induced by Diophantine triples
- Dujella
(Show Context)
Citation Context ...r rank, and we are able to nd several examples with rank 9 and one example with rank 10. We use the sieving method based on Mestre-Nagao sums S(N, E) = ∑ p≤N, p prime ( 1 − ) p − 1 log(p) #E(Fp) (see =-=[10, 12, 6]-=-). For curves with large values of S(N, E), we compute the Selmer rank, as an upper bound for the rank which is easier to compute than the rank itself. We combine these information with the conjectura... |

11 |
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(Show Context)
Citation Context ...r rank, and we are able to nd several examples with rank 9 and one example with rank 10. We use the sieving method based on Mestre-Nagao sums S(N, E) = ∑ p≤N, p prime ( 1 − ) p − 1 log(p) #E(Fp) (see =-=[10, 12, 6]-=-). For curves with large values of S(N, E), we compute the Selmer rank, as an upper bound for the rank which is easier to compute than the rank itself. We combine these information with the conjectura... |

7 | An in nite set of Heron triangles with two rational medians
- Buchholz, Ratbun
- 1997
(Show Context)
Citation Context ...thematical literature on these topics. See for example [2], [8] and [9] and the references given there. The existence of in nitely many rational-sided triangles with two rational medians is proved in =-=[4]-=-. Often these problems are deeply connected with a family of elliptic curves. This is the case for the problem of the parametrization of all the rational-sided triangles having also their three median... |

5 |
Luijk, An elliptic K3 surface associated to Heron triangles
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(Show Context)
Citation Context ...lume of the History of the Theory of Numbers by L. E. Dickson [7] for older results. There exist also a more recent and extensive mathematical literature on these topics. See for example [2], [8] and =-=[9]-=- and the references given there. The existence of in nitely many rational-sided triangles with two rational medians is proved in [4]. Often these problems are deeply connected with a family of ellipti... |

2 |
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- Bremner
(Show Context)
Citation Context ...the second volume of the History of the Theory of Numbers by L. E. Dickson [7] for older results. There exist also a more recent and extensive mathematical literature on these topics. See for example =-=[2]-=-, [8] and [9] and the references given there. The existence of in nitely many rational-sided triangles with two rational medians is proved in [4]. Often these problems are deeply connected with a fami... |

2 |
triangles via elliptic curves
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(Show Context)
Citation Context ...econd volume of the History of the Theory of Numbers by L. E. Dickson [7] for older results. There exist also a more recent and extensive mathematical literature on these topics. See for example [2], =-=[8]-=- and [9] and the references given there. The existence of in nitely many rational-sided triangles with two rational medians is proved in [4]. Often these problems are deeply connected with a family of... |

2 |
An overview of algebraic surfaces
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(Show Context)
Citation Context ...s being the number of irreducible components of the bre. By transforming E into short Weierstrass form y2 = x3 + Cx + D, we nd that deg C = 8 and deg D = 12, which implies that E is a K3 surface (see =-=[11]-=-). Hence, by [1, p. 311], we have that rank NS(E, C) ≤ 20. The numbers ms can be easily determined from Kodaira types of singular bres (see [11, Section 4]), which sELLIPTIC CURVES AND TRIANGLES 3 ar... |

1 | Triangles with three rational medians
- Buchholz
(Show Context)
Citation Context ... a family of elliptic curves. This is the case for the problem of the parametrization of all the rational-sided triangles having also their three medians rational. This problem studied by Buchholz in =-=[3]-=-, is the object of this note. Let us brie y sketch the argument of Buchholz. The medians, ma, mb, mc and the sides a, b, c of a triangle satisfy the following relationships: ⎧ ⎪⎨ m ⎪⎩ 2 a = 2b2 + 2c2 ... |