curve #488

y² + xy + y = x³ + x² − 121779768817463152210722809903374127611030600160x + 15685515258825181630241214599987142280599667999917044892017001532791041

a-invariants: [1, 1, 1, -121779768817463152210722809903374127611030600160, 15685515258825181630241214599987142280599667999917044892017001532791041]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 46280630592701913424698071736504337159840138785939421143334596914282051671126722
discriminant (Δ): 9298758585267493634514928795257927126758939209477780337161389303610313499566171351220390957926145388738891430167969425490514686106528117686272
Faltings height: 26.0024
naive height: 336.8692
primes of bad reduction: 2, 3, 7, 11, 13, 23, 29, 31, 43, 47, 53, 89, 107, 127, 131, 271, 317, 347, 349, 379, 401, 751, 967, 1049, 1721, 2620589, 3222413, 29174072025857701
regulator: 2881538532718811151.846872016209351345832644220272879491749646965549491
submitted by: Matthias Breddin
submitted at: 2026-09-01 19:28:39 UTC
last updated: 2026-09-01 19:28:39 UTC

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

(866871280602900669827229, 749361238882963427243457247908453301)
(92813910499009763172069, 71987477286354434148551569995249829)
(237903664049574060821397, 13363012937768642040652384897378549)
(1238538719886776603145237, 1328438144481888597342688799491158133)
(20219720442843424429796013/121, 1763612415514024659389107429646357519/1331)
(35720887681824294807433, 106681830777951127169905291838393009)
(531556305821754784724073, 318033119387493073854035480881430737)
(-317584502864035818037227, 149430037709953671220424632860105589)
(501103878659494177171701, 283709250684372126767784669938984821)
(9980229851043359249017554327/36481, 371636303067395280960636881379129369882569/6967871)
(42556093513403717423655677/169, 69024694887207049532644828403481128729/2197)
(-3454074126422370405526547/9, 2072923329126651669609910877814867911/27)
(29837567122316228561773242253/34969, 4774017106676815453913595388342953882842399/6539203)
(1099767006275272834927371106997/1274641, 1070066136824834915599440320329425368908046077/1439069689)
(615704516845546471377497723989/1404225, 359152816673264344687724688153249635772111317/1664006625)
(324618986517462653231893339/100, 5816175556439315462801067500762400412493/1000)

Commentary

Exact rank 16, unconditional: rational 2-torsion point, 16 independent points with matching 2-descent upper bound (PARI/GP ellrank). Fiber t=-2794/8565 on branch u=22/17 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 03:27:03 UTC · history