curve #5940

y² + xy = x³ − x² − 6869104207144206575478998651488914x + 221935782110932075310029314306912213579605752302848

a-invariants: [1, -1, 0, -6869104207144206575478998651488914, 221935782110932075310029314306912213579605752302848]
c-invariants: [329717001942921915622991935271467881, -191752515743845311584138818818023532229315661268055221]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 4724989468798958486439099758364047973605979150400585210554533569409780360104561775212290
discriminant (Δ): -534955726264159471005775930405193349007818809439620591349995668149634040334395060166899590205310937500
Faltings height: 18.3233
naive height: 245.3761
primes of bad reduction: 2, 3, 5, 11, 13, 47, 232366635961, 4704013299320244929, 7146314557069327440124910917308024206632464207787469
regulator: 6086406330887864905172000.8294492650507124739559551338840866283050156228103956510755877
submitted by: apkorte
submitted at: 2026-10-25 23:37:42 UTC
last updated: 2026-10-25 23:37:42 UTC

Witness: 24 independent points log in to add more points →

(44284155896547637, -2142045588190986218694131)
(54707718364695037, -3143147912932972150934906)
(73166265667487062, -10537088188031022201161756)
(46528771101969512, -1748192296858834304484756)
(48153277341361262, 1679506406405193043063494)
(178542819510616423/4, -16508419769382525069209173/8)
(56646203422510387, 3820080173912988170376244)
(-43516802888410496, -20939172317822339975627840)
(59942707812696637, 5056175472049960382327494)
(9191890695036262, -12632195291472963896267756)
(13552330718996282, -11460039623889286617057906)
(14174602165129762, -11287910112519659414302256)
(1495755650634426262, -1826575776599767874524776506)
(38550516182799262, 3797344041979216968911494)
(53737899895570642, 2826093914167535672205664)
(48916652778792154, 1723910434057416523284760)
(51715524424327912, -2238166984058463883637156)
(-69490480589856113, 19071147986205683429608369)
(-80496141933604613, -15914994286977337189358756)
(51272870641691912, 2128077214874640984833844)
(103668092966628832, 24979140510158241772652824)
(59286265252052662, 4803669847421074502375494)
(35286971024933137, 4846048902283146915590869)
(56286779553464827, -3691055465299587202732631)

Commentary

Specialization at T = 19/33 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -6195147929170777488267T^8 - 76325725939813122988800T^7 - 882325168895840084876340T^6 + 137101400449928233205328T^5 - 540371179907883674625330T^4 + 1432666851749665397357472T^3 + 819022076707710173673708T^2 - 1550804727906185486991696T - 817873967341793893325067, A6 = 184510578295520929147533147781830T^12 + 3583110577624752524437698479790336T^11 + 53501492723177054560052620493555580T^10 + 169407308640762620085933160597599120T^9 + 698545865865095496350873237321288874T^8 - 1292699670982664072004165953289975552T^7 - 505316963797428638034138121593355512T^6 + 1943998258237962328932351927104831040T^5 + 142849376365152697481163983826326634T^4 - 1559076824193899778858619351522756800T^3 - 52912284260963932942946689331970468T^2 + 820349089697858180287004856848627568T + 284805687406045189533586630147368774. It is a 2-neighbour of the fibration given in curve#3900: its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-25 23:37:43 UTC · history