curve #616

y² = x³ + x² − 3654616687548040226175224389969525481965129225033325684618580x + 2718395254621802741350811985642905251232189668096742750947829890903523661307068960191337600

a-invariants

[0, 1, 0, -3654616687548040226175224389969525481965129225033325684618580, 2718395254621802741350811985642905251232189668096742750947829890903523661307068960191337600]

rank (lower bound)

≥ 28

torsion subgroup

trivial

conductor (N)

13256124609558049820392283796815008531233788836453627781721026792428981264573670000606421009825381988586278050067799482547057785910031781478625480

naive height

429.9885

Faltings height

33.7006

discriminant (Δ)

-68378572047130417487334760056959483812201379955471860174454228220382973263989782625036108150324260545734183213796040881004450495953118417417334995345846279288550615972682669436000000

primes of bad reduction

2, 3, 5, 7, 11, 13, 19, 29, 31, 47, 101, 139, 223, 281, 396572909, 1954266661, 210214586945260699, 1943268833199420515531429803, 70508602410815421111409529277204421106176714702669924666954151

regulator

48251869702606231173181657600401259.42580381629757590568774359959261688595158366917804250518152

submitted by

Mundoamundo

submitted at

2026-09-06 22:22:56 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-11 19:24:48 UTC

Witness: 28 independent points log in to add more points →

(935897991766755716616572751140, -343226767999042843473278614700281241939172000)
(-10470781680719506117491219794270/9, 62716221212848834888037296242195326053501681090/27)
(227036695019678836042148215682370/169, 1061109242806649572268644874026184638712136957630/2197)
(1693499388266893954052853428640, 1177352567008053816459711895476056550148728000)
(44788751794809035458872773945610/49, 129053293116339595815402102479739258522971471250/343)
(-409054752915897992836410552030, -2035899842993988200552255631357856279344091830)
(1072859873146108501143872717670, -179988173760681297543422710275303882754932030)
(1105246015523238671396608840590, 171110260966812044361200686919826533435728950)
(278814799685867769772535102471940/361, 4095323396353410539199400771685001578813205956400/6859)
(14861546287229777235333831466770, -56840204559518450685590196955281190098004704870)
(2768685041619853830949143191265/16, 92549657521862092085775022552719346724754488775/64)
(1121794642435438291537384225410, 174236219712568840570385836127547260563207830)
(1204661261768358540492807957540, -253051526644856194765529656638438198512150400)
(2287582051854822144588470941590, 2515779994218103699716618494523656055454360050)
(-390674221777830302013432154380, -2021517401698397538069458653012480907915324520)
(420592874262989803666711611930255/289, -3411310148686205542707143052332536978536284848110/4913)
(2838667066790334810962461936758, 3901053913862016233995994956548294955219113006)
(-196536108504281571490441305285, -1851774258888449870263133303772873313232278800)
(-2209998265223107554488720430960, -35443089734105609449012387449166043869410600)
(411969863916931176859928494058424231165/74407876, 7918541489490173030849343591239824113792687974154395873475/641842338376)
(11795467096296840343497073349430, 40009370491304845166861936056002513769455653010)
(477627668669764432403263394580910/729, 15312335447169656844831125927176684244480406529550/19683)
(3569917309618927920091683893550, -5930248503481104576054185122924793351587799190)
(99374454679002674419716151839423244790/71048041, 350748359423948267823514611479716455860625045467670352950/598863937589)
(-705165099347678730549372758130, -2223702888429505366205239879398996471229046730)
(2956125090055987585737025653610022537430/3216818089, 67283508445426917430221486779940033466977025267038760515990/182448271553813)
(45726076529293586943265142462438772270/48288601, 109669229709847412197154878584176747346607527140046704370/335557488349)
(29069419154493941014846400607074/25, -25317973010317369512255934503736873663370142318/125)

Commentary

Fiber t = -2057/22382 of a rank-17 elliptic fibration of Elkies' K3 surface (arXiv:2608.25406). Found by a Mestre-Nagao sieve over t = a/b re-scored at 2^20, then an iterated 2-covering point search (fake 2-descent over 3000 cosets of the growing known lattice at level 0, minimal reduced quartic models, ratpoints). Basis LLL-reduced with respect to the canonical height. The 28 points are certified independent by the F2-rank of their images under the 2-descent map at the good primes.

last edited by Mundoamundo at 2026-09-06 22:22:56 UTC · history