curve #627

y² + xy = x³ − 88108880187467390244983554210163381341807515589530x + 318508301422647662201303129477995684907149943554616095388335083180311039076

a-invariants

[1, 0, 0, -88108880187467390244983554210163381341807515589530, 318508301422647662201303129477995684907149943554616095388335083180311039076]

rank (lower bound)

≥ 27

torsion subgroup

trivial

conductor (N)

255417940288246678150583574922944179897578074304448055056272816920964172850303531273517623698377800620421130739930868904355190006

discriminant (Δ)

-49039776566510582000950184566169350233516872079338741934263021194957711940191240379209795043386946144496881210372526345874210490545126624645563629568

Faltings height

27.4712

naive height

356.6227

primes of bad reduction

2, 3, 11, 13, 37, 41, 71, 1327, 338177617721711, 613548173194227342170437609, 1201184027477738426918374049, 642839906786788643584327182240675424261680581201

regulator

346949950424240469110872242008017.60665602071833961253630511512531717909442967316444143473663

submitted by

Roy van Rijn

submitted at

2026-09-07 08:26:36 UTC

later contributions

primes of bad reduction recorded · Roy van Rijn · 2026-09-07 08:41:31 UTC

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

(16097469004602767700993593172/1369, 1526544981653909181574751583621953108322654/50653)
(1300695039941017398789753933300/241081, 51282282412199970104219054203693709137637346/118370771)
(4162833223872219552745428, 4885101409270745785730264964959537814)
(313130609052607174642358136/49, -1390066165132969837087533901562969911302/343)
(134366986997567550298430553012/11881, 35901761887701474745233167797490238907395726/1295029)
(1717272949456986562673046, 13124998594452988457350634269692059808)
(1843000044654584108586613716/361, 9077202959349721082092485072419037787122/6859)
(253299457743765685169080184916/11881, 116771833896779800165700792736341148815073582/1295029)
(5797634587060851598196940, -1599559342333164223532012528244708546)
(56138997367459476846142404, 415089267305336953460571368307154495830)
(5452841945970221154042860, 443282642107355105380261774045682846)
(-10178799591593170191049860, -12678574947413182303741548998689025994)
(-9057430759630493983458077544/1849, 1999675453304493874673558507062555580792022/79507)
(8107127088446819893202401598196/58081, -23032538075944868796363196672219086668545331082/13997521)
(31258487745492027671221682116/5625, 295038224588108438106686422771546295567986/421875)
(72846330166531120399301119099666283210964/13577475527880625, 751697945714151138619291556589337887682039628526413857707462/1582080373886805535515625)
(62527860767159315631662360458611579188/17850379950841, -548289979121304338492694941399088864302583027353346893726/75417337631284650611)
(34995349130732752278795520632/6241, 429608256375519955702492406367708057828846/493039)
(7997019417386492445897114, 11194990712094137994397028205254263050)
(-46509375571534300147302025685256/5583769, 287420731011448645944066716381475134573376965526/13194446147)
(255160329768396550483765140/49, 325321446680351308698455454558100942842/343)
(-4071315001410756096990861021855094290/420563923081, -4433551628100104466419164380644025890724809798938014024/272739489193336229)
(54551696116160854783526750484/30625, 69302677367794867614239236659893541149437002/5359375)
(-19085448473370657258424929/4, -200843555221823854187831481188699674215/8)
(344808198803564498979884333209884/66178225, 504973492092274943810597319437283140211273845082/538359860375)
(-48498631454963984784801577356/10609, 27334758747942388295509476609892072710268602/1092727)
(-1037321004694596017733372, -20218538522161063305165176216692656042)

Commentary

This curve was found prospectively as the specialization t = -1867/270 of an elliptic K3 family with generic Mordell–Weil rank 16. 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 11 independent directions beyond the 16 generic sections.

last edited by Roy van Rijn at 2026-09-07 08:26:36 UTC · history