curve #404

y² + xy = x³ − 19287996132928291523665979159101108205455355568716415x + 983351868730932985116649853601474646887931496490259148783013795470646286030617

a-invariants: [1, 0, 0, -19287996132928291523665979159101108205455355568716415, 983351868730932985116649853601474646887931496490259148783013795470646286030617]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 161556735231476753254583770823345475052543520784292394332312680868697134445526610416074943640483611548766974670305841924180347140444710
naive height: 372.7876
Faltings height: 29.0013
discriminant (Δ): 41505940491909352111741176345891872662987590833015371871277614219536506459943277907639290641022780946037539336622788189039872262087323170328449646922854400000
primes of bad reduction: 2, 3, 5, 7, 13, 19, 29, 2503, 117466975445805874169003, 40102956406526573322891878089, 1301247044069409176793485972536865652985877115046209823816786383362471611
regulator: 1976469480783608859731990660622237.7497697010948215230220644149035445572892995171740078191824
submitted by: NDElkies
submitted at: 2026-08-28 17:06:58 UTC
last updated: 2026-08-28 17:25:54 UTC

Witness: 27 independent points

(4759109245492436204950374, 944280666776535660774757868974298693613)
(55224962092529877761984574, 294275221747047230540208884270266031613)
(94083069928218594952126374, 38287647939482588317402153098629189613)
(64512185924282150488730454, 86771650364837312936060853603981639693)
(-115082804603438448714503226, 1295726020481087443746049546849429940013)
(4349277385750916197740774, 948443617298541523147586285438193735213)
(-96018067638139622573585466, 1396464234395918627024207379268258146093)
(-82935642627004905134963226, 1418645926850988554239990474034411583213)
(-65882272521685879566150426, 1402900070013471122851144686336433248813)
(96849759689848451223648414, 154119467388364918206606534946249028133)
(194026267372675997908490454, 2131976588065035976704540194051184999693)
(104093572471203561663091014, 321711699539274739973847345583256491053)
(243856890385518290371388874, 3283455432415195363131621627105571514613)
(105202974301014643984073574, 344308971938354143263978538057439813613)
(64795699284404949366604074, 74941913960594764916126825367973073313)
(65420750562084501891014502, 38849630479946419673959986447731415213)
(544567444081111599349471974, 12327742217570537365313509413378740991213)
(64911478819199283996282534, 69599120994074300878944398463782747373)
(53586476468025799846101894, 321947581983001725284326399721384354733)
(65604542362910079658749414, 18187129015601816244242865292521401133)
(104583217355769396827671974, 331731422080656625573614711028904136813)
(450219412340337856280753574, 9140999922505833561679568114076731592813)
(51349876615569498923502054, 358211538748101910774992776442920740653)
(105917631778970072538118254, 358684660244163904785731053608290165493)
(95670676707436100993651814, 117125125218363469753775204927896356333)
(124402096049605714281623814, 713524223978548046643207349812974112333)
(98892241676814576116196774, 207491501477298577199402401048710140013)

Commentary

Noam D. Elkies and Zev Klagsbrun, January 2026. The cubic field generated by the x-coordinate of a 2-torsion point has a class group of 2-rank at least 25, the highest known for a real cubic number field.

last edited by NDElkies at 2026-08-28 20:41:45 UTC · history