curve #581

y² + xy = x³ − 31195586637172116547881951428068973404453330675920568110538x + 2124966784708562488315836765911738609695361122641802486121241080222796144460330841728292

a-invariants

[1, 0, 0, -31195586637172116547881951428068973404453330675920568110538, 2124966784708562488315836765911738609695361122641802486121241080222796144460330841728292]

rank (lower bound)

≥ 28

torsion subgroup

trivial

conductor (N)

466327614591319361199238779339055872306377700125464700689888547207305294084079816938021468859497088226788021318593951936213767879940235870192830050

naive height

415.6805

Faltings height

32.4392

discriminant (Δ)

-7748768158332216877292226494555603215074671522141431044476932638738174826615266389895565174417579425321616012318272778853589515218673641602240254513257310517621789803857920000

primes of bad reduction

2, 3, 5, 7, 13, 17, 37, 53, 71, 1031, 14489, 357967, 521471, 42958608745065856058906287721, 17212943356303065337090528956255492482290187424400122903296392557799469283927156193208071

regulator

254628444491746523551642252751128725.8122866799232310111354171026604353394193629578934833936472

submitted by

Mundoamundo

submitted at

2026-09-04 18:51:59 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-11 20:16:54 UTC

Witness: 28 independent points log in to add more points →

(-159065522633956886263184353012, -55339484961019474889384478494469270633947578)
(3251474932223103358412632279292, -5854534689218309041565376010744229121925349506)
(163597356212488826000244645428, -37416392960827925301441245595354873710659738)
(5328725621483666236874230198148/49, -1478615611143913508651502856860081564339553798/343)
(-129362539509530819745293036368, -63211297413463468372835817769871794988060026)
(-203604601343166958260795127732, 6011395746522266980872239898050484488933702)
(124856581334300921950144000082, -13281658213915495278963589979114914013843206)
(49779458521800304727482758469892/361, 144736845000648313609693744393612558793482571666/6859)
(85568011985347537077902605972, 9063198612270563755563596020613896882942054)
(528736977118990117021211823992, -365302465938420105623935182446569862171039386)
(131770216551490968504417544016, -17386639540061155576960732208303154212152570)
(5220357938439865745950476149907572/361, 377153531966061185595164458210667201017761691768706/6859)
(38516740376138867117589814059464/25, 237535955281351808366248559007826645904456300302/125)
(80655315873505416491158140812, 11556915771612434348870978022262140632900654)
(-101376464816543927049788308948, 65158269826808153251942872324120636029332934)
(35152121525281924367023575956, 32738543219155504556074085412291525405744230)
(123222536235976880695799503988, 12326922439389836569841108485441682615424422)
(211021767435771322657978973822, -70277006056745786859366311592514829865872326)
(-188233493779041231286860709324, -36435703285246345848724378050555214601482906)
(6726137882597876422552341032, -43765800168590758973830112271934660246827226)
(127599209075050595448595000262, 14897937658012710977805671206320492113265434)
(115517123154452002685369616512, 7926324454005556925911452058113812082772934)
(19525251925583457469776571902263068/253009, -1693790990947503138206454682894635036539386535799022/127263527)
(114120115397474167761141417032, 7152260685172792870215308234069181724150694)
(24126566717962001573266763013148004/139129, -2279870573733937781492860539029447766964187834320402/51895117)
(5926337432467634351330362509215546393672/86535400561, -448021008212721879952143622266611389481091245473940154756394/25456032247628809)
(105464311026402447245407444532, 2827689771013822400184170362905873734830694)
(46362333532267581787357502497214/289, 173635688329882543638840641706628211033382956872/4913)

Commentary

Fiber t = -22059/24335 of a rank-17 elliptic fibration of Elkies' K3 surface (arXiv:2608.25406): the 2-neighbour fibration that also carries wgxli's #356, #385 and #351. Found by a Mestre-Nagao sieve over t = a/b re-scored at 2^20, followed by an exhaustive GPU sweep of all 131071 cosets of the section lattice modulo 2 with a 2-covering point search on each (fake 2-descent, minimal reduced quartic models). A cheaper 96-coset probe of this fiber had found nothing. Basis LLL-reduced with respect to the canonical height. The 28 points are certified independent by the F2-rank of their images under the 2-descent map at the good primes.

last edited by Mundoamundo at 2026-09-04 18:51:59 UTC · history