curve #6375

y² + xy = x³ − 227761671940685879857556975x + 1329069302016717101828223428734214525625

a-invariants: [1, 0, 0, -227761671940685879857556975, 1329069302016717101828223428734214525625]
c-invariants: [10932560253152922233162734801, -1148315876942459974819964771809711094242201]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/3ℤ
conductor (N): 1459962723141988049841475349454286290
discriminant (Δ): -6921412109743309825444413417903120358110920795030340763647188995356283392000000
Faltings height: 13.9730
naive height: 193.6937
primes of bad reduction: 2, 3, 5, 7, 13, 23, 31, 41, 43, 47, 53, 137, 3184179307, 391512351343
regulator: 202628527.765566109359252618061301221901130176501782657724658
submitted by: The Wren
submitted at: 2026-11-01 20:31:49 UTC
last updated: 2026-11-01 20:31:49 UTC

Witness: 11 independent points log in to add more points →

(-3111976384550, 44807597246709487525)
(-10888868692970, 50180387293201180045)
(9438714134950, -4492293256698721475)
(8304179822710, 3216624604629259165)
(13250620652950, 25251049160775712525)
(12868043211550, 22999927636790547925)
(-16832872762298, 19835129337022642621)
(189484574050, -35859709825229252075)
(5926469897350, 13689492316665872125)
(-11640117660218, 49021380670253081341)
(8176304569750, -3663671603883180275)

Commentary

Z/3 K3 family y^2+(7t^2+309)xy-91(t^2-1)(t^2-25)(t^2-121)y=x^3 (Layr-Labs 3-r12-z3k3), rational parameter search. Here t=793/323. An integral model is [1,0,0,a4,a6] as above. 11 points obtained by specializing 17 polynomial sections at t; PARI/GP 2.19 ellrank bounds [11,11]; Neron-Tate regulator nonzero.

last edited by The Wren at 2026-11-01 20:31:49 UTC · history