curve #4021

y² + xy = x³ − 48740038757035963863101026358782907238x + 182006773533218992289707120399188676242257095333912285092

a-invariants: [1, 0, 0, -48740038757035963863101026358782907238, 182006773533218992289707120399188676242257095333912285092]
c-invariants: [2339521860337726265428849265221579547425, -157253852332701209341816234815405605671453404266332583640625]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 3633958535429087440559985546561230089154484056735382962822667450
discriminant (Δ): -6900302530841110917910003698703812679227275634052442276524987901228496789785529138216717522643475861087235531000000
Faltings height: 20.7394
naive height: 272.6104
primes of bad reduction: 2, 3, 5, 11, 13, 19, 29, 37, 53, 73, 109, 131, 191, 419, 421, 443, 571, 727, 1097, 2953, 3529, 5009, 28751, 110629, 327499
regulator: 3219856947640144759246385344.90301053678868591779305554683779681159337584
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:47:27 UTC
last updated: 2026-10-05 04:47:27 UTC

Witness: 17 independent points log in to add more points →

(-1806061644291199130351251646128656/451234162206529, 169571714107494576465868647290473720852012503858678/9585239260517371400417)
(-86130816697293922620164875063192416/322658302431738025, 2559351168964353144977947901871756049562046597294345858/183279658448669124529664875)
(2979771104776188231371225424139953/422357411384896, 119426299583916214576557366841655796052069089858237/8680009073461223021056)
(40036354399140019906310793517552623/2260814015728324, 7503589192088164172282577127031214506130464455389993/107497225514502632661832)
(-447064099429231477510128369977892/53392585122529, 684245641885084857369282257578022336476041438558/390140847519777221167)
(239630314765492772787269858775125298/38520762373850761, 82660698939733090189074508718580183171255060531028368/7560357126761171104889509)
(-208070183102377858416291408185114942/27321488570521161, 47687126925879329700548170290885218702759075464030592/4516027067913586482284091)
(18331993965549549442591020684114976252/30922928148463762401, 2129223357955058377141953097572133768054838494910362991726/171957418626666522204998196399)
(-40584897028646094144327530934490462258096213/11359483026725480982693604, 674676496351013850662648123498905410640296204266483496778883650829/38285810942505411393579547019801239192)
(5528412851049810258724238772346249257/808635642212823616, 9434528205614372142728573261282907806530050034439084361/727158892818593839103281664)
(14158044262576494085261241509797650112/184621965978555889, 53062297279108763475821328539331011001353312301941011050/79327735488408145660047287)
(34736496674596048759591529696892225703224/2851651916558692603225, 5689175174279612732822611116734764779970298435162229665213898/152280666575681392502623576547875)
(-2921127335752247515430158731602110974/35474700147003094201, 2881758126608392588300022874458287157080585456698353701920/211289589742090159162645899949)
(768916796153252674020628143967616040297300159361638/42226105506851672028392341555201, 20037197560283747454650269044672770707183296694439770961602158128513766907890/274392061634147883371391469416285734444391191551)
(16404330909342087188668557948308706338982148764/2404477552449942539171120689, 1523815808004433428663499598976372521273877781519202317123074701569710/117904692634050371586750367484148695244887)
(252335388488401102935098954267384581848/23646744508347336649, 3405356604922359298541730752571492374455890079486845129018/114989196308696474067532154107)
(12664712288038146169135771222929457079639260038/4966891203718896478160779969, 3017444166510662512335400361398475919108967784959186531413516039540972/350047492308078712629493088781239174417953)

Commentary

Unlisted rational-isogeny neighbor of curve#3485 (source contributor: Steps Unbounded). Source equation and witness provenance are credited to that entry. Exact rank 17: 17 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 04:47:27 UTC · history