curve #4011

y² + xy = x³ + x² − 32055155015767905773301304657x + 2208812813162227288883572183878136873046725

a-invariants: [1, 1, 0, -32055155015767905773301304657, 2208812813162227288883572183878136873046725]
c-invariants: [1538647440756859477118462623561, -1908414270572175917451212043316788646782047045]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 640259658391727630453646333422455570202860388634738
discriminant (Δ): 349671761133100287884290327107470559752805552990970651191086129142650851706718292752
Faltings height: 15.0669
naive height: 208.5254
primes of bad reduction: 2, 3, 7, 13, 23, 73, 157, 193, 211, 229, 277, 307, 367, 373, 421, 457, 701, 773, 1049, 3109, 5381, 22397
regulator: 7192098266884369865560736.6947706830257206588363641454476681165461308
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:45:16 UTC
last updated: 2026-10-05 04:45:16 UTC

Witness: 15 independent points log in to add more points →

(47163406358272170397323409556393273984559529/204005490695679793583686996224, -246462055078564933070840907605433885377199152931929448220416480835/92143092637618869422563332087506636091322368)
(778937138912442160911142283515711999877991946/5520645421128797075259568581025, -9125132231726847533204526222125791218672625519198409355345744556421/12971338158360178067919848177285519078568432625)
(530568928040982603432299931967881397404206/4332813456784332815474876449, -98683010715810118910118829133593544167135072519534888600758727/285203589572140508019610797946098791536207)
(199104510056906951257769837431068855642769/856189185123136435745773824, -67828677457821886839846936435568781291082103015787224640698495/25052703969217502427236832868941223047168)
(55351128993524018033972381773594478734/774825859515222404289, -411801916550918594014397611937560514247559092152641111255/21567815515580679575974110693087)
(166117387661726761554554/1503345529, 7360110766960810795607785655105615/58289216195917)
(413489460703355133595853011/3968044080025, 46271957334109482640625985268792981474/7904323967189399875)
(707025019172936825286513116517256549034/6677175656392535735287729, 731904044521677210199843187954933487614733982342859247485/17253976494255294967738906321758700567)
(31487301854051820631415275460051/271644587649669025, 31846088433576622171208499869006484740101632439/141579894577451986620688625)
(147944892378962946895577913853534/1290552630221255625, 295495673403956855294487370574805253746911838727/1466099954955654654796453125)
(723783814079645109978476712811/6922685127155625, 9163817645672028964384187010680163409672909/575985920762063141296875)
(29008639033327290959311167103148999054/256465222780229573580625, -22557906457530592066517434639101546475336463285263558367/129880132587787459559295628680734375)
(3053622401521179084245267627921175651694/94747553079935463515025, 168739326433171086339512767284596096621084982142274408564863/29164330035909846330770783151293625)
(1248131196227202572264174723387156822033156709325886/10867637490260080394353255806908141641, -7359510673576069367811820378783493923942264220748977418166542893147091844589/35826362423733534943628419184112614238091705814458709189)
(16036898851239355431741429419909186057929/153061829565744601284823296, 39364273752473404609768827442741487475411404569754969719091/1893652782285815706800048423393003065344)

Commentary

Unlisted rational-isogeny neighbor of curve#3652 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 15: 15 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 04:45:16 UTC · history