curve #4014

y² + xy = x³ − 339010402241513092912707774822720x + 143950247887913955080900318015720759906665567142400

a-invariants: [1, 0, 0, -339010402241513092912707774822720, 143950247887913955080900318015720759906665567142400]
c-invariants: [16272499307592628459809973191490561, -124373014175157657214306623726971679249074009798269441]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 4657372890037610572108520563138067741151339367609178987970
discriminant (Δ): -8949269554993736037631396937806297885050382040469746006255730503512388623128990159197766871477785408000
Faltings height: 18.4336
naive height: 244.5102
primes of bad reduction: 2, 3, 5, 11, 19, 23, 31, 37, 43, 59, 79, 103, 113, 257, 271, 601, 1051, 2797, 4919, 81013, 270071, 599413, 1520587
regulator: 72258173110877537981025435.70626995111873144675091127590801776793230679
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:46:07 UTC
last updated: 2026-10-05 04:46:07 UTC

Witness: 16 independent points log in to add more points →

(-4160661923033059697616872127720/648247804625521, 199334399701165889050293375363376990965696075840/16504850009954853405431)
(43451779200604365037878984/3504995209, 2469697187985831673138023932714885452224/207506231358427)
(116188432557910647302599830/16992947449, 26392086914717720427963348369023513164950/2215149650609293)
(-13688041883739600779980385977113999/307385554849293376, 1433407899900241739686118253551076156789681646418283/170421813284796007361114624)
(2300055219250084118238551554189162989/431914074482890000, 3488229132816123965836326171042565219353993330486250587/283854664004081928913000000)
(17031765173994203125513287097675787880/738903454645605806881, 244665709437184724044401031051840092115164871814853272640/20085449044701685751790417628721)
(-31097569985432227092682500/646227241, 114792795269932588244332407954201846480/16427742693461)
(278858110027449129800863747545963155/284667370950105796, 147241913752751061743805119121146781271529523225928465/151882079846329438339689656)
(-5834318724337729913346551912253852/184840383586232929, 882078561069301073038840663102801463176582156383504/79468550512807276699731217)
(63831695052774666708240771619902943950/2505607724034948297241, 1545515623216645979622148906999980497480479609826437770270/125420815063789756764341010257789)
(53675442042337085311768406385274320/408019988090522929, 12703959361621183384120601589281023098659915472328560/260628652265109750510954967)
(3540139462577196624924285185413619052/80514791111117978521, 10569840098808796599071371619187073487622272966353096656/722459499301753656848256661181)
(10283923857917664524455771306770777180/214811372621863894009, 48514055956742044849220424675698959183284806863558718640/3148371013859576686697172820723)
(3518036245016965105949726695579824/454710678912088225, 3651121490968426864824260270995554320402320475139312/306621589488968537212416625)
(-7420351521442584873313995197239275/963559236533323024, 11432706582532843892504542717795629682444184124386685/945839894786989049316488768)
(-105726559615073702798739295316730730500/1943889121196378477521, 104809336409693733197792018849455061514546298885455256080/85705212078745082547955065094681)

Commentary

Unlisted rational-isogeny neighbor of curve#3655 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 16: 16 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 04:46:07 UTC · history