curve #509

y² = x³ − x² − 54243768548784769896807423046465961442957385180x + 4783866325139741100004372428412867476659045211947418691104840226960900

a-invariants: [0, -1, 0, -54243768548784769896807423046465961442957385180, 4783866325139741100004372428412867476659045211947418691104840226960900]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 3018786376446535998275609684836898200968187274821185538667187516489187463931509795160
discriminant (Δ): 328309339512561184929823039121289205321653805939380365592608577363494992384734816174565567831914060926701719794507135024647495552726644000000
Faltings height: 25.7568
naive height: 334.4431
primes of bad reduction: 2, 3, 5, 7, 11, 19, 31, 37, 61, 71, 137, 151, 163, 179, 383, 499, 1087, 2083, 2269, 3331, 7211, 8101, 13633, 124853, 154571, 19575161, 28502407, 204479773
regulator: 10528819855347700003385602.7112245079341167105026905493783189957591991576
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:41:37 UTC
last updated: 2026-09-02 17:33:24 UTC

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

(12948908284061969468667700558785488636/64028947272481, 23395272679586428591994040289181753618866578303745085850/512347406545463918129)
(126920620817053040563788875968659733/1058433555204, 2057600290631114775061162698004273515250728690534979/1088918558460985608)
(1340895516564754455403705980200560/9032031369, 1474360572912478632831540325426494817849696491130/858377165215653)
(51420441978863500784181027826367100/320680966369, 2623836834040354042787624538796141567243689459334490/181597462402201903)
(2017980057769107278571659939112160/2072707729, 88261199731994890720906370465948276119300382530770/94364164778183)
(15272560156828376549023851773063330/406856449609, 13733597130550255324257402891799909688591170315844460/259514606952449477)
(-42380237404833838504520084859037891960/9093350896237969, 61539428302674760440566802301156703659649113740446452882650/867133420995462948863047)
(49170067264690379086597244981572720/422108790601, 1856249851348837061163333540201197547475511595753030/274243659144679099)
(492452493217031601797500104830215407947824/7445728525419495529, 783078784595781009416404028571310024636099887314193956665743270/20317056877293196847270379083)
(1705776007848404430792355724299657740/6615538940761, 1516185804910446149389134242008733836166603710525574690/17015622627824204509)
(21773858050553270907419633334773600/103207345081, 1732440730878809532957745056117865817510766346186170/33156288473376979)
(1475685258394958128330998508387132210069984848273940/243377377056059541872802289, 56646704590172159122039064521417205713067346026584713834774994289734049998350/3796822630370410678999430127804614044087)
(5924958301318129902827866499976941425395090/39924989526634308721, 379715087177825356831906633793675802620229814761494360936438860/252270934699831531231763990519)
(3841969347544663602368115523008310/34188379801, 65477120919287689341314156166747773161669459831500/6321465613584701)
(245093070214777709265279491857708/4779924769, 15277930657015370780252594305939904817352082280534/330469658754353)
(2534636704535988210570337356239931356877607656/22151070891725150005801, 28588864245004191827823581470513137302184675185684799239362553683090/3296796120771340504479153860782549)
(13398909391752369672630579653694402428786637760/156758532098236501571161, 1724317107749622310277718132753388046700905690627669049246928483724550/62065003193967514189101259853890909)

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=11/19779 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 13:41:38 UTC · history