curve #399

y² + xy = x³ − 11226388884183735861962205254536431339112877364056989183905x + 450194265482766125175008332772346224614695308482085725750790650500195411870167825256777

a-invariants: [1, 0, 0, -11226388884183735861962205254536431339112877364056989183905, 450194265482766125175008332772346224614695308482085725750790650500195411870167825256777]
rank (lower bound): ≥ 29
torsion subgroup: trivial
conductor (N): 7869260002718699409432075483535055935787479608995618427154413774038311615923554081451828153361043617864852200874946393629408760548170
naive height: 412.6105
Faltings height: 32.2721
discriminant (Δ): 2996906487408668684946588524064302639661533663533177016944958230332154017535306076077800764570850204532905563066634896986302693073322454776543014199888730107403271667712000000
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 47, 53, 71, 109, 131, 9343, 583635521501, 3259827042004671353977579, 254545367939161636009405376466140757317631648120501095305055675853519863381
regulator: 11455384821273366074699125993730851.9042013976688745225702601768584968853752279173689481644224207
submitted by: NDElkies
submitted at: 2026-08-28 16:30:28 UTC
last updated: 2026-08-28 16:30:28 UTC

Witness: 29 independent points

(54400171629672808491067269834, 683733243665069377617328769153852791177323)
(54590576223440925805659836394, 162162188581665977725388419839811998817803)
(67878910906653582846112316394, 956271977741217393411940565774127637345803)
(68353751150770733667331834602, 1480973375250285703557774097715039058955275)
(13258821971224249955848347114, 17426314198043397608065641842740270809536523)
(15058314789815143943169427434, 16868855427115941345448110239146521182490123)
(-87994025612271782812833910806, 27508475131199786732165018643119570539950603)
(-39482597530265166834847576206, 28842551989091959130597730725025481178386203)
(30060756264956272321110839274, 11827294697372382780722654006811531246326283)
(68968094055378561231138773994, 1996276308631484861617876135736891906862603)
(147563253543962351914762057194, 44796974469607818606793292599945352860744203)
(-44754791034091818941846470806, 29376614782332404811614118876477081899790603)
(85318588789961345755015131594, 10650396013015443793761016518406110125171403)
(-50437471054008123308591934486, 29801260142864496740543945714910689185510923)
(118728151755109041013275200874, 28123696345359819086225851479845154581583243)
(24995795834934493293246181194, 13608778765421393431399588508484634265862603)
(323295789413365857490413285354, 174961938875441210627132018430887666722135563)
(128029048452706374236601972714, 33338723620988452989831616938898826702106123)
(-67120643148016205324232902166, 30022093182855376660406844534506176808461323)
(-86396019557310551788759945806, 27842883695989095434290500295282512787815603)
(54559765912754960193945322794, 311262353775121545606890117479004920600203)
(37143915241991546526909405114, 9189591403421384203780902479054802207914523)
(43224211287076974096438890474, 6760152110940398496835157251292076377513483)
(28451173735585708140185866794, 12402446380925935942428795077476592760784203)
(368628334096277252854361218794, 215414883177893763657194495399053415649525003)
(45544305211237411049076545514, 5776503729941964641303792804930877561190923)
(762085467699210575781557410794, 659161854222131690804394381897629965449621003)
(67532802799252177669626257994, 200561740695543948967853941229517266087403)
(236226831305723988205878716394, 104787466910773536384731693605249282275937803)

Commentary

Noam D. Elkies and Zev Klagsbrun, January 2026

last edited by NDElkies at 2026-08-28 20:43:02 UTC · history