curve #3225

y² + xy = x³ + 22461831362529182584186836961913400462x + 99681757298539603818055720843145922688167002101080034692

a-invariants: [1, 0, 0, 22461831362529182584186836961913400462, 99681757298539603818055720843145922688167002101080034692]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 1518905128450311243438362902929463593569897853087633327924112350
discriminant (Δ): -5017843881318955450676405422574476112355338462466094866462560637485512806652161009187429710387790442567995584000000
Faltings height: 20.6917
naive height: 271.4063
primes of bad reduction: 2, 3, 5, 7, 13, 53, 127, 137, 139, 193, 197, 349, 503, 1901, 3853, 4703, 13633, 34693, 259717, 491423
regulator: 111593161164090631.9987352169250683137101303817490350618010729079811028
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:28 UTC
last updated: 2026-09-26 20:43:28 UTC

Witness: 16 independent points log in to add more points →

(-1358524569007516818, -8164528737748013886919624416)
(-2234784439901270892, -6190582217455428383109349650)
(-1719872229735609948, -7480839738870867968220829026)
(-2421703974282043098, -5575249272923012681190735726)
(4231171480173269172, 16446017312171818470098786814)
(15697285123146446172, 65727903547114136654157542814)
(4753799047628626452, 17716941567651211752862606974)
(8993675336684768628, 32080529334009684664407115662)
(5188680690708476132, 18865863426485978882291779934)
(545331864463205172, 10587402145483296857892164814)
(11407511585843734272, 42899765798045991708660610914)
(-3062673106784277348, -1469945280631395019745063226)
(166035287329211386452, 2140337097540651803430540296574)
(16145622916781092777632/841, 2127123673299157394083356331211286/24389)
(-3075974597614047348, -1219029401147576527512488226)
(16066442368585324293/16, 710513502944173007902940368005/64)

Commentary

Dujella (2009), Aguirre - Peral (2009). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 27). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:43:29 UTC · history