curve #4214

y² + xy = x³ − 348058322861900570216157331003073465370467690x + 3777353138211892644471768263568785185627152160140934324917967304100

a-invariants: [1, 0, 0, -348058322861900570216157331003073465370467690, 3777353138211892644471768263568785185627152160140934324917967304100]
c-invariants: [16706799497371227370375551888147526337782449121, -3263633111415075244823632839922676457222915029689599478018630424416081]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 942939644904828574080077075792115067906593731930478571634760145916894965223947101427947162266137817956890
discriminant (Δ): -3465362751078331218348813958919756738988424519330973721318190120787291282742763596067470683157328350788259522439408269553226315366400000
Faltings height: 24.7070
naive height: 320.1224
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 41, 47, 59, 79, 217754441, 385955269823142789727961, 1808572461127438133859688103976516527025876422170296677229023
regulator: 215580745635293414559483426850.10475154960005974175334551447344729906607728747817145660150353
submitted by: apkorte
submitted at: 2026-10-14 04:04:51 UTC
last updated: 2026-10-14 04:04:51 UTC

Witness: 27 independent points log in to add more points →

(-9364256772650191799720, -2493093555803296785300238674853790)
(12825770080399858494580, -1192929647124812497286791867618490)
(-14240692635183276626060, -2417844150472849589859709061291450)
(-68142897500178114703805/4, -17458963316494254798642044351851045/8)
(5346977160729859835395/4, 14564563617090357873372162603098255/8)
(22378715959869194838580, -2682474750338480771578153312820090)
(4778133314921857653220, -1491097421323012812868995904973930)
(-6640066355018532723620, 2407430118749239785017032758385510)
(24672344755898367144421780/1849, 97662791611095721111852130162650569970/79507)
(6923725150986275137030, -1303611254383554355098362532650240)
(16132504556237239282780, 1536525054260697730025444567963710)
(65348757354451822271380, 16127660534587081574026510003885510)
(6179658115956351596380, -1364720323205887699985443882356890)
(8418468761577934530292, 1201606045811175435809490974170822)
(-16081865537596599778370, -2283765598691680436688427518101990)
(-22431050673495160627340, 546295009529141540728332149783110)
(24351237661691565357400, 3121148297868796031784188788090210)
(-42283523521179196980995/16, -138434192887091383563002263432252535/64)
(1547933697062601961780, -1800636258015505532444953800826490)
(-1069538304893370937100, -2036760022424706505255227163255610)
(16162657786394896744180, 1540779355457823391411695677053510)
(22262357427477509895220, -2657490971940693218749142406639290)
(-20611908029920955695280, 1481389428336586690298979893194570)
(8640486078471729623380, -1189555148722598152578544685208890)
(-22391971859918630038220, 586277730660860968845401015864710)
(-788081766441375048600830/49, -783285462520149019105792665394483370/343)
(1004505328532386247699800, -1006593821715591130743472723231215290)

Commentary

Specialization at T = 2617/2100 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -123856789139706169159947T^8 - 444860715403915335809784T^7 + 386197275608827081416504T^6 + 1527252937315431153244272T^5 - 1375436505233375995520880T^4 + 1122976665942695182127232T^3 - 920913248897305236746496T^2 - 558254638916475632295264T - 66664274811511860464112, A6 = 16359416402689331569871278519529286T^12 + 84307064804680497817942655822018088T^11 + 43582982500411768342190410573462056T^10 - 406313301062273227318175674455353520T^9 - 696574825662812869066591591106450400T^8 + 831815699985675007793533243030026816T^7 + 2339654226324227927870456252492273088T^6 - 2418192605459673378876521710121559264T^5 - 588423094738870081007533655250779760T^4 + 462827873819015983743769059572260800T^3 + 312326563016748800922506015981603904T^2 + 77162055820545612135770412695494272T + 6735862148389763480692013144006784. It is a 2-neighbour of another fibration built here (itself a 2-neighbour of the fibration 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 2026-10-14 04:04:51 UTC · history