curve #3110

y² + xy + y = x³ − x² − 341028288631925619255862529986918076734891191227x + 11958344258090135357107222075346466514765426831863562447599675830871851

a-invariants: [1, -1, 1, -341028288631925619255862529986918076734891191227, 11958344258090135357107222075346466514765426831863562447599675830871851]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 66720045390731263315331475333869520786441558783362426910
discriminant (Δ): 2476571304159078724946965409894372059043504842525010210137431842407518798893739905865264835519620627637020354867897179012747377603031660100000000
Faltings height: 26.3966
naive height: 339.9585
primes of bad reduction: 2, 3, 5, 7, 13, 23, 37, 47, 53, 59, 61, 67, 83, 101, 103, 151, 191, 283, 317, 443, 523, 613, 1063, 1621, 1747, 2383673
regulator: 376943949397.596571954100729819232643387661372222441880385673
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:37:44 UTC
last updated: 2026-09-26 20:37:44 UTC

Witness: 11 independent points log in to add more points →

(1607264533354644592931408979/529, 63248939747686860158997055863001989978158/12167)
(-14411121206539022861796519475040861/41960244964, -2558026381484093992271875843833380013204956137606063/8595220498915688)
(-1166610325036982299259231635043715501/3929739275449, -2298231803695844707602567559180784015028811635297899518/7790146160861253293)
(-24072226885708903145541029310278045/258903898276, -27273866782503634546980909385840204392806660236520283/131737034944183976)
(64442493562810347014388505893416096575039/21602155340098089, -16046310733628736262858634314414269341246661356281923472650638/3175013872011635579193813)
(-1630040612911885937649932528127300879/6677774371321, -4900867314031526769726523384592793561518809925437325344/17256297186131077619)
(4788667894307097249177370308671755122396869062586491/999267354156680035687350841, -328925258243774703525503153345151166286387486699161024062286345640581810260086/31588030524022434688449346131910822906861)
(622290352949480269629266941802607349788232455039476724799356686153351622500320791033231902424819/954914312464268933549691519852335916567493458864019814587511690727975169, -240578001969147287095296145122316132688188290023248671151351598258717845812709423105330167394431993707262570683648949182440268693806260475285678/933139565397850362145846114916132191878903300762120577027450473787093189654078689622271990357050289808391553)
(1032715968535602477072765738038784084568968313289959356479094905348549873571/1753937569992558630551262637084977108834807543644521, -9082400431776181353853389873834859097670866351071708764326717895031156672877646904997337993936531862916360096206/73454971811022512127129236086200015398575706701246297388607105436803125200181)
(244571085934410716175919730500627073706296228451/30777793141218163325041, -120624890426840974149573434536679112766464125357853745780158133764393994/5399533184643216373724332374030839)
(52557468452771355258830979036322393956664093512901277262910310506612201214671161/92603673724713287470462703280239719767602829489403548736, -31185629182855411475993126197836303300893678212458904958507477799292249246131339106756676095868013107635111625836950959/891132584788860846794629156658456449895676754608487338020316649434593512312141033984)

Commentary

2-isogenous image of D-z4old9101112-011, rational kernel x=300388533188292313369803. Source: https://dujella.github.io/tors/z4old9101112.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:37:44 UTC · history