curve #525

y² + xy + y = x³ − 551947310675979391891583294635679131127089152363x + 157099509167570417134702882126736405833398299501722157639537535414016042

a-invariants: [1, 0, 1, -551947310675979391891583294635679131127089152363, 157099509167570417134702882126736405833398299501722157639537535414016042]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 15677359235871344955055989853595029670413032016398269566737332835394059780410056266474
discriminant (Δ): 99630216585965563598177951756063565243354960961544175018243802779459203248698726276377984776784433760383995041597009832908919060277911778018492
Faltings height: 26.2830
naive height: 341.4029
primes of bad reduction: 2, 3, 11, 13, 17, 23, 31, 61, 73, 97, 181, 233, 353, 419, 449, 457, 557, 941, 1531, 2087, 2251, 3307, 162517, 169019, 277919, 1650023, 6171167, 2813681711
regulator: 1030657730556041331249751.87568581700702543101947176445781723721892582014
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:43:11 UTC
last updated: 2026-09-02 13:43:11 UTC

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

(530439955155441135430519, 116501761790458205944436753284846142)
(147642799367561458262364527249/219961, 30781715013266249687602446457234397845570062/103161709)
(4292713922607666691092133/9, 1308267401128264967879610658050145832/27)
(5463996321035297078449925108651489/4242828769, 347617615793941222492603072890983839553137641116670/276365137526353)
(1635396179182666283380456747755523939457/9637285482284329, 247296132293975656074224279159103076292041693245410926215678/946089202952641800091733)
(-3550972949807821414401798268044599851825/15631955523896761, 1016976695486939709544597123019341615919856058442986234412454/1954429305857814408994259)
(3757902777998162381532693903827313602/457797032449, 7255952719857618437627362965015786757071331152137431548/309748676734220543)
(1568730732295574366536366569118684874351390/5285081108186402329, 1693187841847535081963155698694857744783144332174086036111081704/12150033768772599101801764483)
(920391244685888426773268074488265559/2512481776561, 253933189386469422479171843721525436020611482371772422/3982487126873086441)
(-27354945537009754840041229217503/37356544, 93757279514796413906046308627533424873537221215/228323196928)
(7869949428401723243837327921203270232593133/15275163351416550409, 5815615199574752954640868577318134959093495812604530392142982888/59700613609173800649806096923)
(2234690241514954804154317910/6241, 36143584355482685895650701314076213689207/493039)
(3146428045934368502262259464994380997967/7808447628507241, 7517233256909655082309908751988602154012836533924668695446/689996762049493952513461)
(309720011469948671824265909543104592/768040151641, 6690866886412050397741238293190091931774605444202059/673094260054987939)
(30698919756697316645913131328346235/26057062084, 4495129274571179038637104367257999274542556318160129/4206183075723448)
(784766019513518644679924787969857837945/1271500594823449, 10294049364400117661207710645832281312363780251948003207738/45339304260778190551043)
(2331099338891090847585562039759301/377175241, 111770479682686037400004338227220520295680572208728/7325120355461)

Commentary

Exact rank 17, unconditional: rational 2-torsion point, 17 independent points (Neron-Tate height matrix of rank 17; the archived certificate additionally saturates the lattice at all primes up to 97 — the points listed here are the ellrank output, not the saturated basis) with matching 2-descent upper bound 17 (PARI/GP 2.15.4 ellrank, effort 3). Fiber t=773/4404 on branch u=62/29 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found by a Mestre-Nagao residue-table sieve followed by an exact 2-descent-type upper bound and ellrank (Matthias Breddin, 2026-09-02; method: doi:10.5281/zenodo.22242107).

last edited by Matthias Breddin at 2026-09-02 13:43:11 UTC · history