curve #516

y² + xy = x³ − 1642440073507487770141809395441944148053577289699687685x + 805595144091421264235562404674247105134026761576765962899778281565252323218396225

a-invariants: [1, 0, 0, -1642440073507487770141809395441944148053577289699687685, 805595144091421264235562404674247105134026761576765962899778281565252323218396225]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 281137155660058521374013654863701954636215183842935698717211507024645830876359657299507174410
discriminant (Δ): 3201465848473673503881543313334345211840394964094228668724808998319739507096871624539567508802654788245451413523655661131618832483165808164779960129781113536000000
Faltings height: 30.0177
naive height: 386.1209
primes of bad reduction: 2, 3, 5, 11, 17, 29, 37, 43, 79, 83, 107, 467, 641, 653, 1163, 1181, 1409, 1429, 1447, 1913, 3089, 4877, 6229, 10567, 18973, 22039, 41813, 133051, 65209607, 404571467
regulator: 17888354866898702478199081.4235530977106194832040878270999497025273378273
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:42:28 UTC
last updated: 2026-09-02 17:33:25 UTC

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

(5872474515899150534591932962468290/4012009, 315104325309880464891814192811333137171436190595405/8036054027)
(-10545954457564703955614224288274202351903880/22627768172434681, 130496100050170257865776362589319954924586744413955112569899324315/3403788612019025067917171)
(2742877555875798222191265058063324747253066690/3980704753451507449, 8049770733125159581507071598856239826269332632613542681346010577275/7942184124055441099209489293)
(33134378282158163771727527072237163183415963080050/49660696812649112681089, 28778593570113942992798875424725120795229762373525374719851889804116484295/11066727430906122304814074298085313)
(8893123578038470330277397938309790329322884230/13278256840186668009, 118490230348154019831393486642679263299744338073668748889803506866155/48385094920087746471934412277)
(70548140406426984845567428465285398787730/105041447554729, 2513020730081088870549657592303764869842958111169713122705955/1076566959869333543317)
(2451584064696674470004059557222733405/3761614224, 798814788298810786205715723834865237366745938052994145/230707323586368)
(92885893617646015276968412119173095030/4531039969, 893500283262545985192856511634105975276979767147580917455/304997893433297)
(6072282348088971924157150680408595733956898/7635899316278689, 1027907015675133387258601685247397372461558497322000528973605559/667252623478647885297137)
(17455447592433827853274869231781960397963920/25376486495400121, 4588555270593660378753721779034297753556238962909999573824862035/4042474019575887825898669)
(35202968350471474228516159670269185527621838530/64390577465974829521, 4355701308265177205098820548078729286532360485370930492525463984370255/516694073151721785489726502681)
(29811469704773528678909261662608604475270/37793735770921, 212019455505625204942082284159395379328043249094239849195815/232343075443195965781)
(4608272616770187522550288795493904770251675490/4380102940759985209, 142650668603054232210191723064233435962153225505569440222485870661125/9166984057441718080086530323)
(356990400703782082118034496829110528291102501550090/1419154200809975070850369, 34163557366122937793825742091669821695863578848647406119279700582013803521705/1690613686930738791973179259177954847)
(56026379491391850774286764802740801745803680/45593072103267649, 246841148157827553275722057324177120796074931430225529102597107905/9735276359273956173528607)
(465371123128969634782814698541088353589655/53176151671204, 314237730839837833796162879661473215020663992851756424480042255/387771026864450466392)
(-1074776487350171745504871743165339971950/52231942845889, 10936650484667254065126377910365582683307186955123316949051545/377488973681695066463)

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=30311/1877 on branch u=31/17 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 13:42:28 UTC · history