curve #628

y² + xy + y = x³ − x² − 12863021937989197442572245103869706866325173191505x + 17802205930101377427150906610708284800466302835899765794079393280105836097

a-invariants

[1, -1, 1, -12863021937989197442572245103869706866325173191505, 17802205930101377427150906610708284800466302835899765794079393280105836097]

rank (lower bound)

≥ 27

torsion subgroup

trivial

conductor (N)

626331835852237837959942160026415058213286411653252011129135977016421858574282246171665388209084791081551835278559332642950

discriminant (Δ)

-698803764333922225881452682703627988473575385055139144321696385420459618811963213599318605839675657668786479665115883391213066301636816763473920000

Faltings height

27.0467

naive height

350.8540

primes of bad reduction

2, 5, 7, 13, 37, 61, 89, 149, 179, 4919, 526689608973452707, 1297327695885285412767490764778583, 7644533052528481157308027511138991115839138041559182477

regulator

35743860320941141662851642470301.598384221193797810458962096346551288551908088639280181999648

submitted by

Roy van Rijn

submitted at

2026-09-07 08:27:08 UTC

later contributions

primes of bad reduction recorded · Roy van Rijn · 2026-09-07 09:26:36 UTC

Witness: 27 independent points log in to add more points →

(109292241610100662634669071/9, 1097533151962722049118285277817582562300/27)
(54419394496666541094488990150653624521635213812671/2123280991391261848284889, 397713497191692677877530413770879990916860984985217054279985409196414450340/3093935158215003932349501372139339037)
(60394417941969750821807010944831058583399717479/27674315672055276805681, 1623319374408945227391643503689348608540139521799962073643401728859180/4603788203087654604328824332483671)
(626332904887410312858046733360836239/137715952201, -373370405436414713871418685934224507206221457153424380/51106527577743301)
(2770755664029743541006479667727127584231/6610531807001329, 1899058932252875133848442063574121594426335995725731120920660/537470458632372893580233)
(-658287940022183629978772853841615969/165256071289, 162246444190445907533262668350015126757147602280593700/67179402332190413)
(182741504803426930850125819, -2469862470682590485606806240534436822700)
(39159737213043458311891189950938706119/382034011921, -244904769492446442132214193386779384751539557260276027820/236131020394238969)
(4788069264989688594249321645413211/102111025, 330372791195079109619377263131922714074053123657376/1031831907625)
(-1022221666379347038767147094608926505/308106035329, -837101723930488516311073280395725925564818383496662348/171021341348174017)
(28385502512952254483664002191055081528463/2521167517690441, 4564613824014772382332493707430578688687344170821320890379140/126590919578116284161989)
(874094814354439151609352383174463/251254201, -15505629850965130992252449065389440153753533961604/3982630340051)
(-2655410468631381070556401, 5764977546747877343489081969996101460)
(5967066146929259869401434439/1681, -283081410686271489753103299042096075240620/68921)
(1037973308853082544271435878602831/1631109769, 207004540458920408131192794026858052883074724778620/65875630240603)
(4976639057437958715982073138984289150239/2377008193394041, 25597069198137385614292692870078026946248183703276204526300/115890014732411502911411)
(5375549855885789222211484591/5929, 1197926210964982707479990832860747790469700/456533)
(12410996627540248533003735069693579/6051439681, 103035629218394614001328033191855610237783987172460/470747544224671)
(-920892740792601861461777722789043448253/244011485831689, 13527238730859003731474488628234182506690098493817401501340/3811670966649198254363)
(-3236486600900221253509030704234987441/1737574785241, -13608087838410452272362192099221399076571742240337381540/2290420692235914211)
(1899645222030070993070647122789813287/1085478510769, -912545665216743799264013548673706493719978385050879188/1130919897665322647)
(-149392249712752179858946689/49, -1836876806687519387456827717703347474140/343)
(31131854726004058226229613519/14641, -451385099131502081322873175296425234528820/1771561)
(3619513677230778891863526649723/2070721, 2418673712948680708775780088407794856628265260/2979767519)
(78380599592350844377619017364649895/24535149769, 11727998233765513746619321075056937079999595194388268/3843112254366853)
(4545075381747755621258644226857074873/3763751321521, -14658183288143279789464880375175251929013446062925469240/7301824350052279319)
(-51945182919890832231503098220931/20151121, 526146099448064645708226705870192668737072679960/90458382169)

Commentary

This curve was found prospectively as the specialization t = 4286/1881 of an elliptic K3 family with generic Mordell–Weil rank 17. We found 27 independent rational points on the specialized curve, proving rank at least 27 over Q; the exact rank is not currently known. The submitted equation is a globally minimal integral Weierstrass model. In this presentation, the specialization has 10 independent directions beyond the 17 generic sections.

last edited by Roy van Rijn at 2026-09-07 08:27:08 UTC · history