curve #848

y² + xy = x³ − 11164664418280499372340781211856561720993310743403866094775x + 452347706083137132420429396490299265309643996076020970449633416838423618553239978675625

a-invariants: [1, 0, 0, -11164664418280499372340781211856561720993310743403866094775, 452347706083137132420429396490299265309643996076020970449633416838423618553239978675625]
rank (lower bound): ≥ 31
torsion subgroup: trivial
conductor (N): 4495693382842369945911916670933506750043764733213581065233713776814149574753945301199263771044133976706315012174197038516132416533684979553969970
discriminant (Δ): 671865871884746256012805771315369197477767721453025552974409374448729783880032207444786254902764679783453058265217688082537687485026825298358421443648933365862811160576000000
Faltings height: 32.2073
naive height: 412.5939
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 31, 379009, 217030530469, 6688300221028123, 3882334767400026440571512867305327977755492163677255111417251376234703598962635489756621651229698342683
regulator: 36624422283106233203938551318071670929.47430682252589326342671998808687330986090327856056183868428575
submitted by: wgxli
submitted at: 2026-09-22 17:35:47 UTC
last updated: 2026-09-22 17:35:47 UTC

Witness: 31 independent points log in to add more points →

(57218701078468290567741107950, 923568580088759083659433202655335451549325)
(57224022109916449505041926550, 919695112330442121607598807179887867716725)
(57523436975521346388241735000, 678064401069994130106748428957805381187275)
(57546213933925068818878625590, 657013650643863507167242685761715318808085)
(57897813501492133509238909270, 143518287370045689715077112918587166571125)
(64050214586358567076298676550, 100376293409116164998985858754099992056725)
(64684647195444638814916913950, 901438413696317978028864414422577054017725)
(37077068844755007482420269750, 9453298700320956202628601201697519860152725)
(43886164346933538622874233300, 6848208534013988304616349433001515275924725)
(49151359380403322417165601220, 4725662461876341018416726069831682053008405)
(50196540142499562904507577110, 4289525999248650701408874259320467549688565)
(51851457297898121916945494550, 3584696110694168466412919345766145544020725)
(53411427985328368084157120950, 2897951726089820534310832298101275428492725)
(54579236852281001612937072550, 2361049392070935111801315184210079721998725)
(56702668689178319939465772100, 1261523438320548650085592804546481696552725)
(65190430587342162428727232780, 1250684346759681660263665875324840010667845)
(70200200833874728928058290950, 3812794098509529395035321773275013483242725)
(78192176813052471731312592790, 7578030891390922630306934736457079466702325)
(93978290811800058996255822550, 15268277319273297996130082767294664260748725)
(94106417809268577531949553800, 15332620444951331877200659633365984813252475)
(-45233448582802899822412807850, 29407710041179746156943113088789995797252725)
(117668353964875603090021422550, 27709879157007787033767586753075023877988725)
(174532968457358867688433255830, 61808790399703501589994611900411883664171125)
(725866602420262089962623439830, 612204682638379303969788056474269643688011125)
(1605301980898502888351062320166/25, 55910741298205010320876892715259819597973921/125)
(162023228062974589165719289126/25, 2437540134807440647711389546592060377298387361/125)
(56476862152013802903720661787590/9, 424369987092196551252672984334701424261969514455/27)
(47482272603709913602994373432550/729, 23901043601710841191613139988555366932663306175/19683)
(809291070847677074646412602784774/5329, 18490842696643555933591026214888479550472543917309/389017)
(1506462024755672824838351784648550/38809, 67285612330260818405414543012565588649294818824225/7645373)
(3658645971983415072918087269882590935670/65089786129, 25631307625279300828780412245404217970386124985413556317875/16606161865733383)

Commentary

Specialization at T = 1156/90195 of the rank-17 fibration of X1152 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 13930T^2 - 58222T - 116429, p = 22484T^2 + 15779T - 2958, q = 664169T^2 - 1950111T + 11356, D = 4(5453310324T^4 - 7247452517T^3 - 9449920314T^2 + 298734782T - 241579860), E = 104761558671T^4 - 17002430008T^3 - 87813120009T^2 - 9024681002T + 117178968.

last edited by wgxli at 2026-10-04 18:07:38 UTC · history