curve #5161

y² + xy = x³ − 1473721342753431752259542845x + 21710621949432889570623063437116601159537

a-invariants: [1, 0, 0, -1473721342753431752259542845, 21710621949432889570623063437116601159537]
c-invariants: [70738624452164724108458056561, -18757977364310122696955005056754906088924809]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 5430668857396047456406533044904719163125021637089857700586223679970
discriminant (Δ): 1221664636793366424392019216810808381787987683405510515276020788064339430912000000
Faltings height: 14.4222
naive height: 199.2864
primes of bad reduction: 2, 3, 5, 7, 41, 125526913, 52557916018672529579, 13657686673091233425844408014183493
regulator: 131273110306049777153.78807229212216125921354242052413069507947054898403641757352
submitted by: hdeping
submitted at: 2026-10-21 06:26:22 UTC
last updated: 2026-10-21 06:26:22 UTC

Witness: 21 independent points log in to add more points →

(-61788887594251/16, -10583102634372458020273/64)
(-266109842580567886/6241, -41331330597233679975930463/493039)
(-417253195405051/16, 13179570443898262255877/64)
(-125762521080337/4, -1538120717772794356031/8)
(-37168806707686, -158548487234126514457)
(-8000821421225884/289, 998337913841003551216879/4913)
(-18792704479006, 206806421805498421583)
(-37614636792736, -154675887802562423257)
(1207435434546818/49, -6492920936946443944615/343)
(8247771328514, 100582131434644569743)
(-22267393267486, 208531930915539297743)
(-155225878731649/4, -1144314445637302099211/8)
(-528821889825463414/24649, -806695163861584108436026501/3869893)
(-41981592896203054/2209, -21492663978428229432305471/103823)
(24806682290114, -20440407830173737457)
(-1842427235735356/121, 268255674019188764513383/1331)
(-30933334801036, 194161083649795402943)
(20611808255042, 9561249197299809455)
(20471066717714, 10985108905239079343)
(518572310877266/25, 1018368448602662943571/125)
(21132340446914, 2152411598805244943)

Commentary

Fiber 130 of the x1152 K3 fibration (Elkies-Klagsbrun-type family), specialized at u=-5/92. Rank 17 basis from fake 2-descent section lift (mod-p Shioda pairing, det 1152, index 1); extended to 21 independent points via kappa(z*)-covers of the 20 shallowest 2*Lambda_17 coset classes, ratpoints H=1e7, exact halving. det of 21x21 canonical height matrix = 1.31e20 != 0. Bober analytic rank bound (kappa=2.8, GRH) = 21, root number -1, so rank is exactly 21.

last edited by hdeping at 2026-10-21 06:26:22 UTC · history