curve #5716

← home · all curves · JSON ↓

y² = x³ − 240425163794510515066617212406075

a-invariants: [0, 0, 0, 0, -240425163794510515066617212406075]
c-invariants: [0, 207727341518457085017557271518848800]
rank (lower bound): ≥ 8
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 19626543983225348168703445910700
discriminant (Δ): -24971440054586634638549750960968514950706944786274640768781063230000
Faltings height: 11.6112
naive height: 162.6431
primes of bad reduction: 2, 3, 5, 7, 31, 61, 67, 757, 42331
regulator: 59720265217388.1660240816549802497418389049227892695176
submitted by: Maksym Voznyy
submitted at: 2026-10-24 15:54:20 UTC
last updated: 2026-10-24 15:54:20 UTC

Witness: 8 independent points log in to add more points →

(545616858603451819/7590025, -239380575575568120868804372/20910518875)
(7780279794847064008249428815520883214970819199285490089/48584175466700558399468719642459753721203600, 21056827970514873688732431903837204638696958560770505684972366346589687796608289813/338643118581572498915527557076892427490040456531809164356091784000)
(4354107114142721575296706474623187977972292965306379/51762686821501649914669736332778931421225, -221813602106099047289871228134851097663288315568270393012042994462216357471492/11776743508631634079075575837735797139130720791956799519000875)
(2944384320000237869957674529794520536756893955531/43662847982067533597624828043514458169, -2347948595435746774112200675303929015557373884312431086937675316694102296/288514795792278489442808448000881298113654526819467433197)
(3736483063888623490662188776491334378779036001/31661507738193276337344362311318276, 211033885798558750660100582785415098351869840686311860880271230156849/5633747632028180308795248185823271383843473853686024)
(1409360695646805150425361419507192360921551/3419889640503195431037773645809, 1670267416983804947748913108199196444010973912833865847223841426/6324382633783487581355631955703820636022611127)
(27096791632340823528635875657588770257494968618602239/7469250186931287863042605182289996050625, 4460419585426959703589140216382051216214593007250657972254760741041627595187138/645528632104512333192334451533446658678140270779766338609375)
(10672756316555286891729751397658500240022215734592659/78122148160569939759323226210731889387025, 1049322790856059491344748568949012293697868906783858856217582264964333284911952/21835405683653024558644890393704099304067914449308273189546375)

Commentary

Voznyy (2023), A = -5968127242642530, [r, s, t] = [35, 89, 221] for S3 in https://arxiv.org/pdf/math/0403116

last edited by Maksym Voznyy at 2026-10-24 15:54:21 UTC · history