curve #6358

y² + xy = x³ − x² − 68893432577279986x + 35758830844183721039938288

a-invariants: [1, -1, 0, -68893432577279986, 35758830844183721039938288]
c-invariants: [3306884763709439337, -30895629834493753541814203829]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 2092395623846281989575586284587073442270631686578074
discriminant (Δ): -531468488456955625352198916285116654336740448390830796
Faltings height: 8.9904
naive height: 131.2008
primes of bad reduction: 2, 127, 5099, 114413178301, 379335714889, 37224192927208298366821
regulator: 153965067841172654742908963.32400880759496003045124399600471882094861997951
submitted by: The Wren
submitted at: 2026-11-01 13:59:04 UTC
last updated: 2026-11-01 13:59:04 UTC

Witness: 18 independent points log in to add more points →

(114412085815782391/6121654081, 2812367066047895408540383793/478964336951521)
(-1795114389871058573/9672525801, 6176283318338736770353332412/951283240002549)
(-18322696703990176057/77015685289, 132935054961549934885495856784/21373161934347413)
(56711184647526995239/122958227716, 435647633704953414548050803801/43115794381526264)
(886823842841443594983/217695963241, 26361163966428135409979585042576/101572364833022539)
(-1110398152279169391977/2999349450769, 16851805602831736399787968328151/5194462337857152647)
(80220659885385429397543/9080808579844, 22711537088624712135531186409019285/27364453645227943672)
(95301675719644812636391/74725893403396, 29048295732750995492899825652766309/645961559098824029944)
(116331749601825452631587/31815172564036, 39589640986865969860878047149403447/179453289956409673784)
(1816892948150407217707267/1338054187012929, 2420566678428571812758337507438734348/48945250103667036359967)
(-17706313526264337415087689/46596364620571921, 26744191495607197898608317794127418903/10058380312649444780001031)
(67619415412635873431504199/78241935190324801, 545693052619886461337759023464263665583/21885646252551261365872799)
(254579131930523251808963107/3417719622322756, 4061914043562375974387919725054148622705/199804215954030176369096)
(-272448707746093907057825229/861827452435504129, 4075395144272236946555766186183472716503/800074605077193383459706367)
(1592427900100658500643746343/891149455826794129, 63057186345165004809951993791318940726775/841251437522665335337757383)
(-1141687527987941710747390057/2942095526373780841, 10167162291836436934519918453523772142187/5046440469791621754043048139)
(499794302971169429150675987/5672414727024615241, 74454343598501967358085290463945954267140/13509894694287085022239809211)
(3583073001241298502110656679/85722781592889961, 214473661468480167350892502163236456134219/25098304404689720165993291)

Commentary

From the Layr-Labs elliptic-rank-challenge-public repository, submission/board/1-18-mestre-e-man-b.ec. The source attributes this curve to its own search using Mestre's ten-root construction (Mestre 1982), with roots [0,4,9,11,13,17,84,100,143,149]: a GPU Mestre–Nagao search followed by point search on the quartic. The displayed model and all 18 rational points are reproduced from that source. All points were checked exactly on this model, which was independently confirmed minimal. The minimal discriminant was fully factored and every bad-reduction prime proved prime. The source claims rank 18; no matching upper-bound certificate is asserted here. Source at commit 140b9d566eeaefe70e423bca75e3c5f091b0eaab: https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/140b9d566eeaefe70e423bca75e3c5f091b0eaab/submission/board/1-18-mestre-e-man-b.ec

last edited by The Wren at 2026-11-01 13:59:04 UTC · history