A complete diophantine characterization of the rational torsion of an elliptic curve

Tin Flux, Crystal Structure, Magnetic Properties, Structural Relationships U Cr6P4 was prepared from a tin flux in two forms at low (a) and high (ß) temperatures (880 °C and 1000 °C), respectively. The crystal structures o f both m odifications were deter mined from single-crystal data. a-U C r6P4: P6m 2 (N o. 187), a = 698.5(3) pm, c = 350.8(1) pm, Z = 1, R = 0.052 for 18 variable parameters and 410 structure factors; /?-UCr6P4: Pmmn (N o. 59), a = 698.6(1) pm, b = 350.85(4) pm, c = 1196.1(2) pm, Z = 2, R = 0.047 for 21 variables and 656 structure factors. Although the lattice constants o f both m odifications are closely re lated, the two forms can be transformed into each other only by a very sluggish, reconstructive phase transformation. Nevertheless, both structures have very similar coordination polyhedra. The U atom s have 6 P neighbours in trigonal prismatic arrangement. H a lf o f the Cr atoms have tetrahedral, the other half square pyramidal P coordinations. As is typical for phosphides with high metal content, all metal atoms additionally have many metal neighbours. The P at oms are located in trigonal prisms o f metal atom s with two or three additional metal atoms outside the rectangular faces o f the prisms. Both m odifications o f UCr6P4 show relatively high, almost temperature independent paramagnetism, as is frequently observerd for intermetallic phases o f uranium.

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“…This yields part (a). Part (b) is proved in [GST,§2] This lemma suggests we study integer points in the following three semi-algebraic regions:…”

Section: 2mentioning

confidence: 99%

Counting elliptic curves with prescribed torsion

Harron

Snowden

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…In a previous paper of ours [9] we proved the following result, charactering the non-trivial torsion subgroups by means of the (non-)existence of solution for a system of diophantine equations:…”

Section: Z/nzmentioning

confidence: 94%

“…Please note that, for each case, the first steps of our arguments can be found elsewhere ( [2,9] for instance), but for the sake of completeness we will include the full details.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%