curve #327

y² + xy = x³ − 262245552282515254065846883236370270x + 44358788730180391915082790816530673754979831118248900

a-invariants: [1, 0, 0, -262245552282515254065846883236370270, 44358788730180391915082790816530673754979831118248900]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 20104103494321955336468350880904230636927014428004269314605639576859835351810
naive height: 256.2774
Faltings height: 19.3479
discriminant (Δ): 304214595275859552600848995003180962782409909159288730912914252576839066152627207520825504275130200320000000
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 37, 41, 47, 53, 67, 181, 497515775378019961778973281628748896957770970223246090192831
regulator: 35127044543168812914.5419850411931715282153585140589294332311437217394962193394
submitted by: 7fff-zip
submitted at: 2026-08-24 19:15:18 UTC
last updated: 2026-08-24 19:15:18 UTC

Witness: 20 independent points

(13300346987625972273564260/106151809, -126921815258878457468294861989774951090/1093682088127)
(-33688166634582792040/529, -3000122613725233862935284941990/12167)
(-2100623373568360, -211918990647899567038818670)
(-12122782898776960, -218027872156845422981493070)
(-6685372494009737455/64, -136054172245105804800590924645/512)
(-129747757360393960, -276044063302146916637961070)
(30115526577658613405/16, 159602682789982699897778101235/64)
(1890335766860344187650160/619369, 2564222251576741738531089760348085990/487443403)
(176958571250987687380/49, -2331337958147719608024398483210/343)
(167120388140571616136900/104329, -65122064428459284963995738972655890/33698267)
(24789987240215142247704860/173896969, 227836860152742171266578411399684967590/2293179330203)
(154070655135638090, 87245274648428173106770880)
(3122757635543465935700/18769, 187769141159479781998025697024490/2571353)
(905614790834583884540/5041, -19748287388268200023640363787170/357911)
(26886519463147669646060/157609, -4237594015421216481362153199426310/62570773)
(272527088373601593437120/7921, 142254704961815703015440521806003070/704969)
(57613267467222487340/289, -6660857143461319173200479070/4913)
(129682215442330089596960/1682209, -342208045127931493996874749664883310/2181825073)
(140944368806311259946001913660/1011700023889, -104412502605487029724292177925693901668383390/1017601270128344537)
(-33926193151924324, -230687523263038492005199918)

Commentary

New in this search: Mestre quartic family with roots [0,158,863,928,3783,3850], T=47779/20. 20 independent rational points certified exactly by Cremona-Brumer quadratic characters; quartic point box [100000000,50000] plus an integral-x search on the minimal model through 10^12.

last edited by 7fff-zip at 2026-08-24 19:16:54 UTC · history