Given an ordinary elliptic curve over a finite field located in the floor of its volcano of ℓ-isogenies, we present an efficient procedure to take an ascending path from the floor to the level of stability and back to the floor. As an application for regular volcanoes, we give an algorithm to compute all the vertices of their craters. In order to do this, we make use of the structure and generators of the ℓ-Sylow subgroups of the elliptic curves in the volcanoes. of ordinary elliptic curves over F q with group order N = q + 1 − t, |t| ≤ 2 √ q, can be represented as a directed graph, whose vertices are the isomorphism classes and its arcs represent the ℓ-isogenies between curves in two vertices. It is worth remarking that if two vertices are connected by an arc, the corresponding dual ℓ-isogeny is represented as an arc in the other direction.Each connected component of this graph is called volcano of ℓ-isogenies due to its peculiar shape. Indeed, it consists of a cycle that can be reduced to one point, called crater, where from its vertices hang ℓ + 1 − m complete ℓ-ary trees being m the number of horizontal ℓ-isogenies. Then, the vertices can be stratified into levels in such a way that the curves in each level have the same endomorphism ring. The bottom level is called the floor of the volcano.Knowing the cardinality of an elliptic curve, Kohel [9] and recently Bisson and Sutherland [1] describe algorithms to determine its endomorphism ring taking advantage of the relationship between the levels of its volcano and the endomorphism rings at those levels. When the cardinality is unknown, Fouquet and Morain [6] give an algorithm to determine the height (or depth) of a volcano using exhaustive search over several paths on the volcano to detect the crater and the floor levels. As a consequence, they obtain computational simplifications for the SEA algorithm, since they extend the moduli ℓ in the algorithm to prime powers ℓ s .In [15], Miret et al. showed the relationship between the levels of a volcano of ℓ-isogenies and the ℓ-Sylow subgroups of the curves. All curves in a fixed level have the same ℓ-Sylow subgroup. At the floor, the ℓ-Sylow subgroup is cyclic. When ascending by the volcanoside, that is, by the levels which are between the floor and the crater, the ℓ-Sylow subgroup structure is becoming balanced. The first level, if it exists, where the ℓ-Sylow subgroup is balanced, is called stability level. If this level does not exist, the stability level is the crater of the volcano. Recently, Ionica and Joux [7] have developed a method to decide whether the isogeny with kernel a subgroup generated by a point of order ℓ is an ascending, horizontal or descending ℓ-isogeny using a symmetric pairing over the ℓ-Sylow subgroup of a curve [8].Volcanoes of ℓ-isogenies have also been used by Sutherland [20] to compute the Hilbert class polynomials. Another application has been provided by Bröker, Lauter and Sutherland [2] in order to compute modular polynomials. To reach these goals, in both works, it is necessary to ...
Punta de Gavilanes (Mazarrón Bay, Spain) was occupied from third millennium cal BC to the first century BC. Overall, the archaeobotanical remains suggest that agriculture and gathering coexisted in the site since the beginning of the occupation of the site. Depending on the sea level variation, the site passed from island in the middle Holocene to peninsula in the late Holocene. Seeds from this archaeological site associated with radiocarbon dates of ≈ 4200 and ≈ 1500 cal years BP include specimens of eight species of shrubs and trees, of which a winter flowering shrub (Coronilla talaverae Lahora and Sánchez-Gómez), is an endangered species that no longer occurs on Punta de Gavilanes area. The seeds of this endemic Coronilla species are associated with materials dated ≈ 3900 cal years BP in a Bronze Age cultural context. The vanished species was presumably locally exterminated by human alteration of its natural habitat or because of environmental changes. The species of Coronilla identified from Punta de Gavilanes is known from relatively distant seashore areas. This endangered species could be effectively preserved by reintroduction to areas that it occupied prior to human alteration, in particular Punta de los Gavilanes.
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