curve #3987

y² + xy + y = x³ − x² − 1793852737626810609215688731787137030904219096230x + 832229384741549758195947071912886816794661793604782292906505570058265397

a-invariants: [1, -1, 1, -1793852737626810609215688731787137030904219096230, 832229384741549758195947071912886816794661793604782292906505570058265397]
c-invariants: [86104931406086909242353059125782577483402516619025, -719046188416698991081297882660542882319496199085765835049622137219016517625]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1097108044453769307704470853301745036152608455549132667550
discriminant (Δ): 70231272483696067720627399053055454118099559294743153994968537650390513887501605443406693368836654448903659395193667227782384322099443507600000000
Faltings height: 26.7188
naive height: 344.9390
primes of bad reduction: 2, 3, 5, 7, 11, 13, 31, 43, 47, 71, 109, 139, 167, 181, 227, 271, 523, 947, 1061, 1063, 11953, 4943153, 588848633
regulator: 288443385334928656152.30947907030789717825272427704425729232812
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:40:04 UTC
last updated: 2026-10-05 04:40:04 UTC

Witness: 12 independent points log in to add more points →

(3281863077864858598767454585313805569/3360149291041, 671659342332070829460883502635450506963216364318831845/6159392221077816911)
(-525258446587894515009782370314174151/1005276943225, 1285598605119671106038163531791472073956201666806445673/1007925847970397875)
(13009814654158865574890813666442975267/23788221118489, -14089724016900461438956393746735066873392393262681536275/116022695260965414013)
(-81033233492919750136498140442776978715917511/270174766463323957921, -5146950066588254995720030663710091045909149944215621318414864939995/4440860971810919715808572248719)
(3809320519976728034503574833937222766852051/6756350274754861225, -135286858520406513859801845048081638288822382481462012602932229/17561767992942020888840192875)
(-6724212050408828785850191989544531276779/4418159201485225, -56525967047627242621044122535763204240873272879485880780411/293671662230933769552875)
(247425017983165740382355120863215180473425207797/5244244217067708230569, -123024632029499097808144795585922720578382239871778794960833786281854495/379773241935813462742219222208747)
(-740656292060759081184280332919365518755796425891/1188712876906001327802601, -1693810097832531377652208410260699394054850015964330099881711111130864145/1296031182956615367555874333423167349)
(48497036059054204068501428708491880610869698282769/156215155443570618756120841, -1078726621120931336323385639146661463254321436397114149036065236477672944345/1952471700992596398616714653848491244389)
(5380319778634861125029936894310759905553888679986110021/14952583561138132779331650087025, -27930094887965596168251694592963449693888668995462881779121252602387398768068981809/57819503383385437166516141367950583244849327625)
(5252039869525281657957310900921884564023198481/5371894581238612380409, -44801601035786113693176470534904933786959871159875927418060843313585/393723395776536257616118226888077)
(15161956867426857100882170821668777938025811806612851/8687173378237296169861124089, -1406604335822926647376650186078539516961674695146303101673937755004245769744365/809688057782855652104075785048469907000563)

Commentary

Unlisted rational-isogeny neighbor of curve#3706 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 12: 12 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:40:04 UTC · history