curve #536

y² + xy = x³ − 206303970027367271328508799096648072090692207211688044345x + 1146363239944536854766045164851842125688606849699680362406173842916627534783917180537

a-invariants: [1, 0, 0, -206303970027367271328508799096648072090692207211688044345, 1146363239944536854766045164851842125688606849699680362406173842916627534783917180537]
rank (lower bound): ≥ 28
torsion subgroup: trivial
conductor (N): 331813103309752395293076343879380660629593091089724287829911665027271920509189837373148747464882001851031838749812089067693777612337474890
naive height: 400.6306
Faltings height: 31.2223
discriminant (Δ): -5755688297098923807531501381668506468625850328093169926347617518570650445310491523414976701942646541257835907983144635013998458176758644138436409143554885132700672000000
primes of bad reduction: 2, 3, 5, 13, 19, 23, 31, 41, 43, 67, 89, 101, 59148985348820199420909634084892671259798282531359330875298642860971577589100666301642440718613748549224035298593314750657
regulator: 9073810305849315921726777993269402.943289780073712273989518895385220216077369839514054633464450
submitted by: Mundoamundo
submitted at: 2026-09-02 20:42:01 UTC
last updated: 2026-09-02 20:42:01 UTC

Witness: 28 independent points log in to add more points →

(9254240400395094762944162614, -172391379078727292154199856339364993761957)
(8121454688350964358402222874, -80932174707589793069306500591231179062477)
(-208882841085425658836848166, -1090617937385180970117674843866650342894797)
(8323184490065876376517970254, 76479877485683301154837722448754789356363)
(-133072612653785254263061093574/9, -26512872981920808493255603779924852696173039/27)
(-9010415026854009228053227286, -1507883898103001585246908182561866495322557)
(7896691901555454406168921522, -98306839151840601693202513782526770230165)
(9057437861913312410883684064, 144307022039002944848817130320662457854293)
(8248426090340695182659655456394, -23689504281249720358279129824489725508905378717)
(719952708894615921403379745184, 610759424785809249313911132492662005475391253)
(76642641534389518610201778214, -20869614304667074519084610650298602976777057)
(616388515415041235947581394, 1009670257848401575187937906659924320172323)
(77587130550668094214752216892, -21264899589076930274825928931887073053567695)
(8584249622991474849270342304, -89253379635998003411969580227283620185067)
(7217160071433902836467982474, 182640534266443987638242515101269854466323)
(9263970317014706572518077914, -173821560126419947582063570532663937036557)
(46886506118053990197343419532, -9723486086325088273924851738288274316389055)
(143256728596911066930161986956436462234/18006687721, 224225770538284291058184807662225329835359386510451468847/2416299418593269)
(350982254088586591844623923167196138154/37280272561, 1413462612762737574193639087690089744053906290036318271563/7198112306350441)
(13874782420394737522631592333958776922/321162241, 49130014601644919578350286278192892973792887442745557555/5755548520961)
(6492523150832946800750093914, 283916074742228894732414425727168691054643)
(3634725206767104834001389852893097746821/60659749264, 213344490194830883002708727739728203202630858408279332675759/14940010965729088)
(71680341800314120686243664876/9, -2484844083137260408405033062605641639329439/27)
(52875698266472426377980696597011833411642/1567927204561, 11199704993949737766047305452931123677062989077187732443790715/1963309839807942809)
(144972450972495025980236597853464554/16152361, -8645807719190935885147038751315734383751832094969263/64916338859)
(3649841632191848899748197203947686/11449, -220281220693925230007516657669552827566311243993251/1225043)
(2964454990814679927929465767402/361, 530767624333648174250894668381580984761121905/6859)
(1008314155705351191436734058251670378/11580409, -999516944505358956893542094655519511856449236041078615/39408131827)

Commentary

Fiber t = -26437/7707 of Elkies' rank-17 elliptic K3 fibration (arXiv:2608.25406). Found by a Mestre-Nagao sieve over t = a/b and an iterated 'bootstrap' 2-covering point search (fake 2-descent over random cosets of the growing known lattice); 17 points are specializations of the generic sections, 11 were found on the coverings. Search run by Claude (Anthropic) with Seth Lupo, 2026-09-02.

last edited by Mundoamundo at 2026-09-02 20:42:02 UTC · history