curve #5602

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y² = x³ − 1638128631922146994111643063037629976075

a-invariants: [0, 0, 0, 0, -1638128631922146994111643063037629976075]
c-invariants: [0, 1415343137980735002912459606464512299328800]
rank (lower bound): ≥ 9
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 6552514527688587976446572252150519904300
discriminant (Δ): -1159257059160389977510121862000138186683206870337432400918051794637342991279230000
Faltings height: 14.2336
naive height: 194.1119
primes of bad reduction: 2, 3, 5, 11, 17, 23, 31, 37, 41, 83, 89, 367, 947
regulator: 1105782775070404.7220452802704147189196815097465622569840
submitted by: Maksym Voznyy
submitted at: 2026-10-23 22:05:51 UTC
last updated: 2026-10-23 22:05:51 UTC

Witness: 9 independent points log in to add more points →

(798418572138011737532330202122356290666143810839129/716072462893955406308031154836056196, 22560342534343558044607516626243765447142313755995642258933832562413236232067/605948143731245545665043125011386508372374755641178744)
(20193942054479620384282315525560232807829923443389814862497684325943062723262293634458134486375481/362923144561294193147656881371741876918651251799924100923513189821108810177197328164, 90314390437951159832487028429999822609145121807134832472678426889576986334939114019085894580447302772749880658625587221069023281628326773267838221/218636163383593791466606523094707395128022410614083191548282602469342779153217727210347506848412071447263346204165458026467288)
(1205481471743731948688603031309829455721/9182352645443843648176144, -41839216357988059227074659860884588174365926754887028655669/27824729534696123556994701666114830272)
(2127732729990784922125923948382219/26219461784163592681, -97996141274384980504360958725665482587043538464128/134256604378902350533109884171)
(33223166208833923175949272796699994257313646827657736579407145929/2631920621615383792158047166794932306508402122985796, 2608785422242843138752393049700033071394089268673098174339566625246456363163515485545278024983333/135023443722087062770842225662612559209633788446625926282267731588940654790344)
(970318670608940395809960849762721809022840038321929357302095280411857984739/7996562163013475706709695574288096214998260578021128804753441, 30211507178892231352550631401227501853063778333861407362741564568634190249701714153278896259400236381296656510188/22612833057963170530433565004917567484497876786241043146619880434244446324883291713114396239)
(1264361014825839309948624551543857991020276404736223534570301366961/54061498784529822901034405620295833042880222775745636, 1327550384758494501528947367106109258767884759630886586555045389020185086917537796505672338590444809/12569908683039119634640616580983844654008193555246743261879780680245505890670616)
(319315663121000728814303200823041288142920378201/21369036139031884270892863204800576, 128738429018667832622980589147871330282427179644598121244118533095523549/3123758136194644178308038522885510653623435372813824)
(32107911669029541679473359085868289329639727466799/2717432913622611209472239187923930689, 15124398715367389345788412450957968452667443884553750754337971954062814418/4479589798337410412170272827420866339411859179417910113)

Commentary

Voznyy (2023), A = -15578373575175474030, [r, s, t] = [74, 267, 415] for S2 in https://arxiv.org/pdf/math/0403116

last edited by Maksym Voznyy at 2026-10-23 22:19:56 UTC · history