curve #329

y² + xy + y = x³ + x² − 16575816647974516238335754592413x + 25996833738682985808506465614332304868873612531

a-invariants: [1, 1, 1, -16575816647974516238335754592413, 25996833738682985808506465614332304868873612531]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 5828132005406968037770167747623110174516223438517658736647919417759236327050
naive height: 227.2718
Faltings height: 16.7059
discriminant (Δ): -483551538222869881998543760521303109946455701683075824214473152684439136068904191524864000000
primes of bad reduction: 2, 3, 5, 7, 13, 426969377685492163939206428397297448682507211613015292062118638663680317
regulator: 361042181815555790913.512234502128084081606251076656609566935927822537576899009
submitted by: 7fff-zip
submitted at: 2026-08-24 19:17:37 UTC
last updated: 2026-08-24 19:17:37 UTC

Witness: 20 independent points

(444226461849009989, -296065949383519030947500162)
(285316058376951015/4, -152158536738701474857563469/8)
(119948689485761235/4, -41178103173156145618059139/8)
(-2600566741471505035/1849, 17149942716363680891855292674/79507)
(-1047756134226289785/2401, 21419596332849183712776397108/117649)
(6606816863449317922415/3087049, 97697485068772354526551877032276/5423945093)
(154331319852866995221615/67683529, 4172064379956401387412803727850156/556832393083)
(1685893605317875332297095/683142769, 195684949471868387133315863502268516/17855302553353)
(2639383489477751, 25174459314881264520260)
(5473774756283255, 315072903426741416712772)
(31829441734598323671591/5564881, 4515340891762404777388628704359772/13127554279)
(31803966046307764695/2209, 172850329505832067944832604156/103823)
(70762535775867854329731315/2761817809, 588160406423278154128372064271293658764/145141811316377)
(5046012642789497815/2209, 750376678488493308648266236/103823)
(21941200566705822735/3481, 85161072051876567529822795628/205379)
(12520520947667379008285415/19737159121, 347847058560646447485602806930667208548/2772853747750169)
(229390313215511415, 109848634975296087867636292)
(14389525428316005, 1663415579976555228142972)
(559636820498128952795015/78961, -418657956462293055019116465358819228/22188041)
(398951755860475915147535/1388382121, 7542574120221595966297871610549980692/51732506210581)

Commentary

New in this search: Mestre quartic family with roots [0,115,206,234,731,784], T=22841/32. 20 independent rational points certified exactly by Cremona-Brumer quadratic characters; quartic point box [100000000,50000].

last edited by 7fff-zip at 2026-08-24 19:17:38 UTC · history