curve #5977

y² + xy = x³ − 1633647840776585531395838671273440275x + 803886275691497853798665378509005054182227807131950625

a-invariants: [1, 0, 0, -1633647840776585531395838671273440275, 803886275691497853798665378509005054182227807131950625]
c-invariants: [78415096357276105507000256221125133201, -694557742197454145799669531567694525073945209693693039801]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 767184420646850251170860010157875054011394223246364552948651118273090703689670790970
discriminant (Δ): -139888682619763220612450894292989216393432516999733029204363337268870503246745744979581765959004569600000000
Faltings height: 19.5391
naive height: 261.7657
primes of bad reduction: 2, 3, 5, 11, 13, 19, 31, 103552570267, 1735950052382287979273, 153545253088410405622926766477427549138230337
regulator: 760251331933429139419822.69037519637391911437433857392585813934697752734526278626219250
submitted by: apkorte
submitted at: 2026-10-27 14:31:46 UTC
last updated: 2026-10-27 14:31:46 UTC

Witness: 24 independent points log in to add more points →

(743599313353793050, 16511264290104476181584725)
(8495517444887661550, -24496512429776026475846032775)
(742481158198209550, -15724611343066478897176775)
(740005009581002956, 14523064768616481660998377)
(284940208240805050, -601272715345822752124057775)
(739408348326150670, 14360902602360304600701625)
(532537664207966350, 291432694505461940809927225)
(3288230417267841550, 5566500985043723349042967225)
(675413466361589590, -92794950765328870434220415)
(749931443249323840, 22841053776416534775014575)
(5097336572158161550, 11176757784634231022412871225)
(765407926610984590, -43508190281342120287519175)
(633133374497097550, 152858762551463917681567225)
(1991042870531546050, 2333284914763901743414961725)
(748934726573265550, -21692855022976005305272775)
(5385057956307164770, 12172390129992312429684007105)
(1120730982991665550, -616998506385120498660472775)
(719320917903633550, 31017714461906299720999225)
(19076984175307750, 879049568784570792518062225)
(695984668016678350, -63432491148577287592072775)
(733979388463665550, 15363063200625585683527225)
(3251256515738684575/4, -913316429448887700886759075/8)
(743076512363133700, -16126733074268213484220925)
(-1064307285561422450, 1156285215768913546906567225)

Commentary

Specialization at T = -488/111 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -36741768965072427329307T^8 - 175652477872961149894896T^7 + 69292619378708331563604T^6 + 676777688422838786893536T^5 - 656364464021699831415330T^4 + 2063898400836725040248640T^3 - 1358206219455103158319372T^2 - 1679620299616449035353360T - 380387730244665052932907, A6 = 2972887747061499967014816310768950T^12 + 24971701851338630896585499476986576T^11 + 40958623108017859037060407448604996T^10 - 185504583684287485388498717737301760T^9 - 545461524648606440857833405079456086T^8 + 682919486625522225231856567512795600T^7 + 2337615182727970756165249196632581528T^6 - 2526741785379638456321704985966591088T^5 - 1756453725043689246538106067241758246T^4 + 628926877625456418301614685250524320T^3 + 1296733620825853845986136477111316036T^2 + 591132703001133261198186526777781680T + 89026717350891792884482981951773094. It is a 2-neighbour of the fibration given in curve#4187 (itself a 2-neighbour of the fibration in curve#3900): its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-27 14:31:46 UTC · history