curve #4284

y² + xy = x³ − 482936255142077762832857361256918863x + 127748997600391189830907215398551042650777655852973817

a-invariants: [1, 0, 0, -482936255142077762832857361256918863, 127748997600391189830907215398551042650777655852973817]
c-invariants: [23180940246819732615977153340332105425, -110375133926737988048675244474577699774237624667467536025]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 5084368801484321084038637605177674753072542398692670741298716815560457210693766644242550
discriminant (Δ): 158418358193986670589235343102890435042304256561129708288520797479038682826409418310332667625982065889280000
Faltings height: 19.3786
naive height: 258.1092
primes of bad reduction: 2, 3, 5, 11, 13, 17, 23, 37, 43, 15668111, 4331441340349, 5614534240395316891486875292563840312435116723892689583941
regulator: 2884900720586821506487715.0560373321060476777849979816623368124311165378545498586194585
submitted by: apkorte
submitted at: 2026-10-16 17:08:46 UTC
last updated: 2026-10-16 17:08:46 UTC

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

(344057674201221132, 48157399098850269161793069)
(365221709912945466, -9272727065214527762623995)
(358228012862510982, -26798126085313444840165731)
(44308333318879130238/121, -3595826457299508497388743553/1331)
(-593849757367745298, 452896218372721533670067589)
(332739710674683799872438/76729, -189607948845658569459258308755551783/21253933)
(14362813898718137082, -54370050415087173186962887551)
(104186148332274342, 280293853333664194922710269)
(25498591871836918374/49, 45186160866659103797561729115/343)
(117955693242776112, 269119116218593348853051259)
(101549964321040332348/961, -8314653621554600923044262065645/29791)
(454856680685737482, 46793232138984628201034769)
(442197204084439722, -25738204060786035059259471)
(766826159429999382, -456433753372001561496866451)
(-776116664521669242, -187255193680771002544594275)
(445320529230327942, 31610907198586956899957469)
(14020906512619276422, -52437276260893388843176258851)
(-141698540750454634842/289, -2440064602209834753766152281523/4913)
(-197679403769866074, -464209867870406560028365731)
(215742420022958982, 183305338660649105999650269)
(338757341192117208, -55003772172991585312712925)
(2905852298448271923/4, 3203033664467926181656646847/8)
(-799202118288826308, 56948057655928049500811469)
(1957289949861155848/9, -4905345766710433456859180927/27)

Commentary

Specialization at T = 253/102 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -159593418515992752T^8 + 66594122607180400704T^7 - 9577916877180584429856T^6 + 378921666915696440595168T^5 - 8319401481873790592668080T^4 + 116272680309493432400708928T^3 - 728840755857976029224439616T^2 - 494050735965340585862790156T - 1390993922871848594371288527, A6 = 40790046358717726655041920T^12 - 19761746514541051225237666560T^11 + 3565895854811577447633633290880T^10 - 339557681946125505951450862243200T^9 + 23467317247680868700633154198453120T^8 - 886973909798626440691615033339785600T^7 + 17462193623232977878169289412672255680T^6 - 168744348646923648755279986373234058480T^5 + 648041970284178877886227382317646179620T^4 - 1598469487562293877807341226375296120200T^3 + 21038628236849997090209222042624669117476T^2 + 9643147208228838672635695562458112132532T + 18608807243853177902952156937051966196946. 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 2026-10-16 17:08:46 UTC · history