curve #5860

y² + xy + y = x³ − x² − 3050388586371204296436134103372092x + 68105537899105130922361819361903558197932382164991

a-invariants: [1, -1, 1, -3050388586371204296436134103372092, 68105537899105130922361819361903558197932382164991]
c-invariants: [146418652145817806228934436961860401, -58843184744826832458036677272504546252808611862180649]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 5482964318949809323506534297906311454985709025939332441422020958716999870
discriminant (Δ): -187231239596558651158985340741408448221817390546917881155267876252246254153741612633666583314432000000
Faltings height: 18.1878
naive height: 243.0134
primes of bad reduction: 2, 3, 5, 7, 11, 17, 29, 31, 53, 61, 3533, 501299, 19516397507, 463263129281127072835382699302210927197049
regulator: 202021601423944573140.2219872813764486081984152259663107810825255795924273839011463
submitted by: apkorte
submitted at: 2026-10-25 15:31:57 UTC
last updated: 2026-10-25 15:31:57 UTC

Witness: 22 independent points log in to add more points →

(22280172491336821, 3346993752882960252677189)
(38514004792877941, -2784202011290630736939451)
(24254092314882421, 2896354245067963701470789)
(38009048625970021, 2659770229364895518214389)
(33842145807916661, 1906027775968250797769029)
(39189222653279221, 2958005957532022303973189)
(3044091488468410309/529, 86670427334203004397834014563/12167)
(-59284641212065579, -6370310709279556376265211)
(-63908993392154179, -1423201175548020025233811)
(53126819437405621, -7483071649880788856798011)
(38788054211112421, 2853776014128007131290789)
(3924995362671829, 7496215290240195914493221)
(-63723338948447579, 1930652759207177162900789)
(33551302154999921, -1878667076165577609784211)
(87050530589074421, -21499271137904781589009211)
(149117781550021789/9, 126966512871637593861263303/27)
(-3955263040445579, 8950349189614086745310789)
(-62875604089516619, 3366277256398750472574149)
(23315814419144539, -3107789365632988257453571)
(358837120424869789/9, -84807212918864651961760697/27)
(14934160374016021, 5087366559113391336403589)
(30213074567020021, -1877050135585160999735611)

Commentary

Specialization at T = -3/41 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -77688200046122639977675T^8 - 345051545551894951495160T^7 - 26575925672984725510240T^6 + 1397379174658735482223200T^5 + 924444105192629673167160T^4 - 552617251497682425745200T^3 - 3294538182497011117811760T^2 - 2836363778933196694377600T - 685235941086191704065792, A6 = 8343511083085250123050296186761350T^12 + 55798071641305709411620408773693480T^11 + 63992696069821126283929842778614576T^10 - 287548516348103372029428932872522080T^9 - 549777946830148798846989863404282920T^8 + 702321721822785932402284527824455920T^7 + 1394771712241954046774251387009862160T^6 - 1833587512252380081925577530665112512T^5 - 1898687314820271014676246427139553840T^4 + 2184401546825882198833306353761222400T^3 + 3375300416127856002328341391809073920T^2 + 1517503055739234059908068559049779200T + 234697979821554858808091582382514176. 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 15:31:57 UTC · history