curve #515

y² + xy = x³ − 1547542866227232895053631088398968680612813307225x + 723364271955589209744266212018571526651261866391858242759877284200212921

a-invariants: [1, 0, 0, -1547542866227232895053631088398968680612813307225, 723364271955589209744266212018571526651261866391858242759877284200212921]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 71504239127973300880966109863118283531248692050825111681599812381652833211716013826
discriminant (Δ): 11149833330883655057294100720239245622463300761506897035376857753438192206974460580710806177679164191158940834121380769261497161437078744422715392
Faltings height: 26.6121
naive height: 344.4959
primes of bad reduction: 2, 3, 7, 11, 13, 17, 19, 29, 41, 43, 71, 89, 101, 157, 233, 419, 431, 1373, 1489, 2243, 5867, 7103, 37529, 55009, 3786919, 1286020398220673174161
regulator: 588801609565891721391078.732075450745452885892768142880957946014021560455
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:42:24 UTC
last updated: 2026-09-02 13:42:24 UTC

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

(2134086120281460864504615402405531489/2643902016064, 48148218736640233823336654612253225234052258611483117/4299005829336192512)
(9625830693395214085237779997218/9006001, 14562559957680533577238316386277541746952158619/27027009001)
(5853403402092197864414341619046627498/6966363735769, 2345171968661810616726327728440819636664917571003317497/18386929881460133603)
(-55170110527911587732533978467962586/66357244801, 20478774056569302610917392249098549399922100426835365/17093559903492799)
(70971199336520428510635551573298/86136961, 69844674559146735257619892810644176941322913483/799437135041)
(8905851505959445958120243224915841274/10911625725625, 2276465858606899266780434884155400883340525642195643593/36044100468813921875)
(750081415113176027169535299761611401150/1198678937095249, 112890644931050932931548195265209361231161763968334113473/41500593835412296448807)
(20648704291817694941418809933191567329762/25294267746308089, 256351910896141124409275391969396286077923308068379515922407/4022843772630558885548813)
(2707015916409867894085411861594594602/2566554590209, 2114491033255623464409277013564792845877670449330315229/4111741081580557823)
(608679630024355243332626286415429386/678160485025, 133829408476909832513512755911956988029087484235362451/558468550220512625)
(1366393654474900058885295283953299491506/14938090906719289, 1393537415020805539716394352961381404249449593848912039214297/1825755623808046166285587)
(-740611176434284625050048044526204303321344/517051938804918169, 13063715974301489675764177439021760054564060851723820612274815/371793040814190370121253053)
(68307895556107052650676356084272966189606/81125775651605161, 3037528595698784250878682388679124136279317428635001510286039/23106719548267481226709909)
(-39407730953702469614737723880991721508925414/28649018760067174225, 76582354173672213755261651057419217225048465528537018313386422821/153343237334778357949052318375)
(38812780389114554934455942192795220/9409582009, 7331667319075941140264470281801597331486076210854617/912757683619027)
(327040435295581792078536018251759300427234/398681135212008289, 19321084783904606577224295717201077209577806263726443967554643/251732059724063876378208913)
(-349911680318269039975771864667263026/1350636784561, 1651437414090946138892785851048604438931210489355826275/1569668201276472809)

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 2). Fiber t=-8631/24595 on branch u=29/13 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 13:42:24 UTC · history