curve #6274

y² + xy = x³ − 52985771304720796247589477212025238x + 4785582739583461021675138580438266946107452599636004

a-invariants: [1, 0, 0, -52985771304720796247589477212025238, 4785582739583461021675138580438266946107452599636004]
c-invariants: [2543317022626598219884294906177211425, -4134743487000110326542295267438559971263281405351324593]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 12079050599388792997553643082553593819574810818
discriminant (Δ): -373122405758427398148345731160332326786195546704476603384033993153830621350549458406541519927788760465408
Faltings height: 18.8534
naive height: 251.5180
primes of bad reduction: 2, 7, 13, 23, 37, 61, 67, 101, 349, 373, 563, 788268211, 3270311898781134307
regulator: 725913829766315.07924905774027253424830146759526629214699064298
submitted by: Maksym Voznyy
submitted at: 2026-10-31 23:35:38 UTC
last updated: 2026-10-31 23:35:38 UTC

Witness: 12 independent points log in to add more points →

(8839478258221293287388190086166916/63044346619386889, 168134491166345274216952788889871036447618234525866/15829573809042800667288037)
(110078742939144285604004876/533379025, 633009226017990638785807628725835828214/12318388582375)
(88943025043239659963239244/779135569, 325910500429721932799389088533410083082/21748011137497)
(24911055265561082358390780/910047889, 1590331494569999519077206419224594180326/27453414667463)
(11503114232260523524675244/78340201, -9074134133729759406224450321555444322/693389119051)
(72932732028303737057701116/587238289, -156303155622768623400651155510512315254/14230545457337)
(105971641744033279617176457580/162475904889, -32593855147358287557938442512675059773903826/65491275170372787)
(-518360932820866915901028763668020/7926393783908209, -63004246290036657203281383710199201882626352946454/705689192720301668008073)
(-366234237513854792519897116/4382042809, 26947851810940875397865293687065821499314/290078087827373)
(3709742285582753513612816484364/12400588973809, 5472960219349186870654268394222325679026714518/43668016840052781623)
(162402399689824662156609820844/1217502594025, -12829595597055269357962515130997757303389982/1343398449760155125)
(12341835637118909293528975065676/149460967528929, -57005572411743608245289664438056429163444123194/1827223550030421761967)

Commentary

Voznyy (2026), T = 4741/1924 for the rank-5 fibration of X163 given by y^2 + (9T^2 - 6T + 244)xy + (13T - 90)(22T - 15)(5T + 6)(4T + 15)y = x^3 mentioned in the curve#4645

last edited by Maksym Voznyy at 2026-10-31 23:37:13 UTC · history