curve #482

y² + xy + y = x³ − x² − 578325186603791398522254226911284910743894333320137932x + 165663567373383964303604617378519352971283772210218024456350513024362673656396239

a-invariants: [1, -1, 1, -578325186603791398522254226911284910743894333320137932, 165663567373383964303604617378519352971283772210218024456350513024362673656396239]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 693388196377898328468926745954415305693424993093406004824100867790323941987867548180169417707290
discriminant (Δ): 523317492936369614307068381771829104616573633705456347502700979685996867589584640927466267722534643920024571888036810244540085248838214667263735513764070400000000
Faltings height: 29.8149
naive height: 382.9895
primes of bad reduction: 2, 3, 5, 7, 13, 17, 23, 37, 53, 61, 83, 89, 127, 151, 211, 337, 419, 587, 673, 1031, 1231, 2269, 6229, 30631, 32579, 33797, 5297759, 5705093, 59371564977849024911083
regulator: 50765749126560690738.84059239803420460925236068056577305815932278621631
submitted by: Matthias Breddin
submitted at: 2026-09-01 19:28:15 UTC
last updated: 2026-09-01 19:28:15 UTC

Witness: 16 independent points log in to add more points →

(809515092541967941718136597, 15099262509653558353555003721799947618101)
(-852747653681017412832573003, 6223279017945511679406039866082156438901)
(30222198675172580753555368484757/78961, 13709308142027746022522232834435957063527329901/22188041)
(504256587536299622367637413, 1502989644611148844965048095854869323653)
(535695397238094697279766517, 3096077922818629568411741080879273953301)
(642119663286090425140003797, 7685497482799719534105976848756619730101)
(360972897710921637904887477, 1984725358592149594600196275971892520821)
(668800808126486807996012397, 8833467738957341198841580557846078067501)
(2257728088463685880622440917, 101825151544314607272702568559802386565301)
(738335663949270640913225933073/2209, 323426453726594807285094744492107422459806623/103823)
(-10697228076736515691528799403/16, 1018923334834875300943691070060516921679639/64)
(13783104989845197131650554679587/19321, 28942388196136411144732783844962148325092045269/2685619)
(-1551182800578102852063681512147/1849, 617882387669644753183223201538726518834186607/79507)
(-305760208818297932235124203, 17717423280050542804358520891743945378101)
(-63851249249720382364549514534627/157609, 1142607852056463517156203946713344675523108986073/62570773)
(65281936272504730485602740327/81, 10888032394802026581466746140177423170880429/729)

Commentary

Exact rank 16, unconditional: rational 2-torsion point, 16 independent points with matching 2-descent upper bound (PARI/GP ellrank). Fiber t=5165/10512 on branch u=19/13 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found in an exhaustively certified slice search by Matthias Breddin. Preprint (proof, measurements, decided slices): https://doi.org/10.5281/zenodo.22242107 — data and code incl. the per-fiber ellrank certificate and witness points for this curve: https://doi.org/10.5281/zenodo.22242109 (Breddin, 2 Sep 2026).

last edited by Matthias Breddin at 2026-09-02 03:26:59 UTC · history