Download PDF by Nigel Boston, Harris Nover (auth.), Florian Hess, Sebastian: Algorithmic Number Theory: 7th International Symposium,

By Nigel Boston, Harris Nover (auth.), Florian Hess, Sebastian Pauli, Michael Pohst (eds.)

ISBN-10: 3540360751

ISBN-13: 9783540360759

This publication constitutes the refereed complaints of the seventh foreign Algorithmic quantity thought Symposium, ANTS 2006, held in Berlin, Germany in July 2006.

The 37 revised complete papers offered including four invited papers have been conscientiously reviewed and chosen for inclusion within the booklet. The papers are geared up in topical sections on algebraic quantity conception, analytic and trouble-free quantity thought, lattices, curves and forms over fields of attribute 0, curves over finite fields and functions, and discrete logarithms.

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Additional resources for Algorithmic Number Theory: 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006. Proceedings

Example text

In most cases of positive rank, searching for points suffices to find the expected number of independent generators, following which we apply a saturation procedure to obtain the full Mordell-Weil group. The exceptional cases are those for which we were not able to determine a bound on the index, on account of the bound on the difference between the logarithmic and canonical heights being rather large. This situation should improve after full implementation of the improved height bound algorithm described elsewhere in this volume: see [11].

35. Here P = (a/c2 , b/c3 ) where h(P a= −13632833703140681033503023679128670529558218420063432397971439281876168936925608099278686103768271165751 437633556213041024136275990157472508801182302454436678900455860307034813576105868447511602833327656978462 242557413116494486538310447476190358439933060717111176029723557330999410077664104893597013481236052075987 42554713521099294186837422237009896297109549762937178684101535289410605736729335307780613198224770325365111 296070756137349249522158278253743039282375024853516001988744749085116423499171358836518920399114139315005, b= 776845386159678589635077615346492181601035042768002014396646962333772688446303892162606526955979081249211 185106671917236143678971202347339963247386055808925185619325909681380265508543158979491984235466881248491 978341526711100575326744746030922470291782156359389005809065313914236892470866399096616908015986267206085 816145609347461468770147859622405813347969542380216159923828490925517451952455079424426512616714569247069 065790676549942365146817589522964032348349807255751358289869629122053879780510640219504941970766697032823 589255263953926885142009701275092664710953135501372398976396568319085695054751879368605289437600720585853 465424006259176930980665902501637183477157293942231705607887213321716750749368884791336280387610317598902 0330254326477036682714837827401377115084796691, c= 113966855669333292896328833690552943933212422262287285858336471843279644076647486592460242089049033370292 485250756121056680073078113806049657487759641390843477809887412203584409641844116068236428572188929747 7694986150009319617653662693006650248126059704441347.

After that, the order is determined by our algorithm for finding isogenies. In the tables in [7], for N ≤ 200 the Antwerp codes were given alongside the new ones. When the tabulation reached N = 1728, where there are for the first time more than 26 isogeny classes (there are 28), something new was required. Without sufficient thought for “future-proofing” we simply followed the sequence A, B, . . , Z by AA, BB, . . , ZZ and then (at level N = 4800 which has 72 rational newforms) AAA, BBB, . . and so on.

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Algorithmic Number Theory: 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006. Proceedings by Nigel Boston, Harris Nover (auth.), Florian Hess, Sebastian Pauli, Michael Pohst (eds.)

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