curve #578

y² + xy = x³ + x² − 1066071946332339100860892x + 372888637161286783309817106150962080

a-invariants: [1, 1, 0, -1066071946332339100860892, 372888637161286783309817106150962080]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 110950909706648386027354584054003218623680117123767799415377945462 ★ record for rank ≥ 21
discriminant (Δ): 17474605846665310266071202941394451872517553379301959211863682296952843632 ★ record for rank ≥ 21
Faltings height: 12.7822 ★ record for rank ≥ 21
naive height: 177.5917 ★ record for rank ≥ 21
primes of bad reduction: 2, 3, 13, 29, 103, 14863288607, 14043772972837, 45014284458994007, 50681794094157328259
regulator: 37117101300010566666.846194974001281678469799609939146074267010474601308284544946
submitted by: wgxli
submitted at: 2026-09-04 18:34:23 UTC
last updated: 2026-09-04 18:34:23 UTC

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

(61858971848437/169, 391254774284499868465/2197)
(73574306667424843/323761, 69507492283158722616094808/184220009)
(-416943626680647083695069976/394218443854225, 4409120454478378562063858518599877386512/7827181578526786850375)
(4314807029600292799/1014049, -8717223840680811112728601588/1021147343)
(442240803098693385028/1059567601, -874201507847005934797419017120/34489984980151)
(4023179527744616264/4687225, 3047785763409394855542944632/10147842125)
(-114655070868852/121, 971052417988300612100/1331)
(-125913479099603/289, -4267975874652115647715/4913)
(2201613806751601430797/8529630736, -267047159103931299088873191669143/787762576254016)
(621765897644415875044/510262921, 10831727384433862078056394501088/11526329122469)
(715283474918641960/908209, -127764856714507684733275120/865523177)
(78448879153189416/323761, 66114251689761930118877320/184220009)
(-782571968688109681570001/1786911256516, 2076589533177278693621972648413226351/2388660769792789064)
(3987553327522614003625087/18490404202209, -31101796404084696532661942241276038644/79509607118496203823)
(517055477051800264/1456849, 348288045006906969532928392/1758416743)
(-137449703296960336613/274001809, 4009383620762187785684480897669/4535551944377)
(1130887581308278652979854407/550669253793649, 33807995626001559903812749887666461376004/12922193589075818655607)
(8729299821502529631624839119/9342095656503289, 396258677302115013294241233685316926338652/902955743447123307206413)
(3665121207599248393254841/3004009971264, 4907418868939931541104633230501330889/5206574114274534912)
(322996487033265730147/168636196, 5068834466174763452209085296061/2189909641256)
(2733550206698179/1225, 129376547831411504601272/42875)

Commentary

Specialization at T = 7/26 of the rank-16 fibration of X948 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 2056204T^2 - 19826992T + 730903, p = -1352(15493T - 925), q = 4336938T^2 - 705587T + 21815, D = -2028(142509543284T^3 + 23186595908T^2 - 2342619577T - 2236095), E = 12(8320656394176T^4 - 4943881953560T^3 + 753235556749T^2 - 40844263700T + 858131775).

last edited by wgxli at 2026-09-17 17:35:51 UTC · history