curve #5676

y² + xy + y = x³ − x² − 13682291025223558218655701168182547x + 629446945747442712028819106850370312740204421703475

a-invariants: [1, -1, 1, -13682291025223558218655701168182547, 629446945747442712028819106850370312740204421703475]
c-invariants: [656749969210730794495473656072762241, -543842161125790500237524846870431374977905168024372545]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 51067301631369846866482631889409436113279841426348090994970974832467991732615078884934278443806
discriminant (Δ): -7230646201520917981104421352051119667461560559312899315769985671203854492520603840079495730157509869568
Faltings height: 18.5206
naive height: 247.4610
primes of bad reduction: 2, 29, 149, 7229, 97671961349, 8369133874567228077277470697712414255723669352613065342160208844200735450783
regulator: 246987248692668316421901793.56250519033321547208987073467939584744427329451042855800138
submitted by: apkorte
submitted at: 2026-10-24 06:15:27 UTC
last updated: 2026-10-24 06:15:27 UTC

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

(102971965631609771, -17674544954332180507165934)
(130489849102225387, -32649286471722596913279678)
(165071669342415163, -53561828489151869999127678)
(5835547273726419291, 14094050102144854885026683202)
(131759146953982371, -33377852596107527756585638)
(-33276343968731241, -32371224413417365164509290)
(139528371732378423, 37904362902604905292226514)
(64902411122250559, 3850089015752807617228270)
(100674893471942987, -16503560582842479877644478)
(-1043856610148474669/9, -691601600920467027722709130/27)
(5595493435304893795/121, 13015115100008779503264485414/1331)
(-36657590090590497, -32889923440829322502256686)
(947948636398512395, -916238100154513759820092606)
(91097801716532987, 11790827371389631002035522)
(101001490061949125, 16669186065720058699903774)
(96085844061960859, 14208589026020623458185250)
(114301737850666491, 23640571514183449690691906)
(56957988956292187, -5908890340667040026226078)
(849016639550624110691/11449, 5825884703512736457189579893350/1225043)
(-3936540709755733, 26138990788673674327234882)
(259407495657394451635/2601, 2126121080410986429006474362134/132651)
(53605988105562683, -7073647482099282639215998)
(-104996117203466773, -30141960676528473771946798)
(347817530583299892651/5041, -1333260491005760089931728237938/357911)

Commentary

Specialization at T = 129/376 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -25474223654459729695152T^8 + 93424157542535492313792T^7 + 249549530991798533589216T^6 - 697407234931421362933344T^5 - 624235669206688152877680T^4 - 597548430366277034810016T^3 - 3503195429735821784150304T^2 + 3443481263994739372563948T - 932157668798019061834527, A6 = 1558176592694379366370572668882304T^12 - 8556444096428535116807109681029376T^11 - 10909694949700311770505743225764224T^10 + 132365209573836556400929013899470720T^9 - 65939622169833360804191934552439680T^8 - 921173694379955547822888188003154432T^7 + 724370040233725908757397970764697408T^6 + 3685802165670854989077288042679139952T^5 - 921815011359411216952545051496976700T^4 - 3541405794655636698734958615621659160T^3 + 4046594122834363561792364624080217124T^2 - 1844492709789748579332256815007671156T + 328955182757185562525000064586958946. 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 2026-10-24 06:15:27 UTC · history