curve #6012

y² = x³ + x² − 8034446327973196958712694600214763821x + 8768067103930889794448627575577589700581037986660490479

a-invariants: [0, 1, 0, -8034446327973196958712694600214763821, 8768067103930889794448627575577589700581037986660490479]
c-invariants: [385653423742713454018209340810308663424, -7575609977796288784717534767755318225411272865336515754368]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 3137286081662052951736824389966369871339603060
discriminant (Δ): -18626685254328311914143810766405568877304075267592872372004638589820948744752347192470397250246603277406636800
Faltings height: 19.9410
naive height: 266.5446
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 31, 43, 251, 313, 1669, 111853039, 266840807022416443
regulator: 2536455792934374.9998154082436497579633575076777518072398858634
submitted by: Maksym Voznyy
submitted at: 2026-10-28 02:18:27 UTC
last updated: 2026-10-28 02:18:27 UTC

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

(3679106932009049260613728818947506909/2259441490912289161, -179296227430770070242356317278480781039553282913730590/3396265624429026669112303909)
(100218441800874715593683665123381/19573968649, 1003278262027562879471494906184509534324805094930/2738535231775643)
(5471640919457564786613216169/839608576, 371502168014208077780298194169869042580885/24328498098176)
(219397935853563571472846898086783461/149682675064733209, 21716252547687719385058979815364531191561160352234690/57910498596696306087735677)
(1302702600340800350023021619726701/782727916425625, 1734899413213191962229397797908044776808280293074/21898594168016727671875)
(1378944915973238297082466/769129, -239730016646097512431784416184891235/674526133)
(-39565978369743147014230691471/12472422400, -2127044529624221216670382602912193591195767/1392920133632000)
(20767650197060885581309309/12602500, -2490145525992744124789374032391075627/44738875000)
(408794450539375437912621070141/5055494404, 261212588637503460548881412377635677640428085/359455763113208)
(1679281815620119625401/1369, -44109500136619274490017230035690/50653)
(2766374744244254035006415696568608989/60560511549926116, 4592496529502406488866735143193616046848560664285251605/14903363781793561552896664)
(-1439712138781744102424819/1216609, 5470472522014909398842743616305990410/1341919727)

Commentary

Voznyy (2026), T = -10272/3553 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 2026-10-28 02:19:43 UTC · history