curve #4118

y² + xy + y = x³ − 173719528525019397613189688778644156848077689219115646293x + 655283790102850006567977331426970882151340170628667781827093572298216118282111782608

a-invariants: [1, 0, 1, -173719528525019397613189688778644156848077689219115646293, 655283790102850006567977331426970882151340170628667781827093572298216118282111782608]
c-invariants: [8338537369200931085433105061374919528707729082517551022041, -566165194648862405674732414365410648232559304051318621090671225758720319819520906706589]
rank (lower bound): ≥ 22
torsion subgroup: ℤ/2ℤ
conductor (N): 2035380499394781053410535073119755867113480721020446687608912956074864478189031910388141364236880110 ☆ record for ℤ/2ℤ torsion, rank ≥ 22
discriminant (Δ): 150026346875966862017364938255318645641599737286555156517681565010052465804753076067620968066820782997435911180330404521388478424637661808904812672601214271499627506250000 ☆ record for ℤ/2ℤ torsion, rank ≥ 22
Faltings height: 31.3629 ☆ record for ℤ/2ℤ torsion, rank ≥ 22
naive height: 400.1047 ☆ record for ℤ/2ℤ torsion, rank ≥ 22
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 29, 41, 59, 61, 103, 109, 163, 167, 173, 191, 199, 307, 353, 797, 1459, 1549, 11059, 20389, 72227, 508531, 5809451, 117291869, 61056459470068873
regulator: 2331918887459416048117563.440644624864126571131197485594214075931269180369794431634
submitted by: wgxli
submitted at: 2026-10-09 17:51:08 UTC
last updated: 2026-10-09 17:51:08 UTC

Witness: 22 independent points log in to add more points →

(191117042808107853266177519, 788726827375293859582996359378326600734890)
(4149949109450523098556090152, 76336970407916205764964138831599312936712)
(4196649478671770612751480119, 12435960133123625871021049286010484781190)
(10602022632913948480268079044, 72134153544336476464025697668463672817215)
(10619284801749662092402530119, 89638137807819322987250604007802407068690)
(1613366449382728480010992319, 615800329225946464263658485253887854287290)
(2241830780646673410911794619, 526403845693493557536478107899321889054690)
(2779564317805939663184591759, 440334009035000136135646317685033168692810)
(4114054819215760234326732119, 101115496382680566329755381325150091054690)
(-1715465733414899949788849557, 973778915894739579299440131056261770090458)
(10672780528256875837410121184, 130119481381752918654548073278680600782960)
(10878754273380083716898731994, 230006223868147999631757656660604591016815)
(11079137694383090107941380204, 300924182507223035392205252027626197863220)
(11094451502510873834604506774, 305846431584865575626110795901645784385530)
(12075234229654685099761023119, 564167681424666903737611356303532116687690)
(13482589727422206091761395324, 874049691843638270039403921419630339419620)
(-13727281106163625058036081881, 673227903469164898472176538902563997385190)
(14985724870693344218925970394, 1190522652383654084680336178440977353436015)
(15464406033143522269721946119, 1291159936129504100913673029994565395217190)
(47537688649719546136735031144, 9991208207415038582886686900624682674232015)
(207937535137955266739027362771/9, 80886100302921380960466895545457705390868230/27)
(1252641227049992627355702427511/529, 6170522012895601096698950228233833300484332810/12167)

Commentary

specialization at T = -97053/246469 of the rank-9 fibration of X1008 given by: y^2 + (288708T^2 + 90128T + 5816)xy = x(x + (506T - 173)(27T - 97)(73T - 12)(5093T + 1977))(x + 4(4779T - 1945)(T - 48)(59T - 17)(199T + 80)).

last edited by wgxli at 2026-10-09 17:51:08 UTC · history