curve #5748

y² = x³ − 40334979939844477688927

a-invariants: [0, 0, 0, 0, -40334979939844477688927]
c-invariants: [0, 34849422668025628723232928]
rank (lower bound): ≥ 13
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 24393495144904469512782813122768868609422244
discriminant (Δ): -702825382114987575602298411693216642374673694128
Faltings height: 7.8598
naive height: 117.6262
primes of bad reduction: 2, 3, 7, 140983, 1714441, 486517463
regulator: 23954588389211488506.62523971304230226376068152351228482036150580
submitted by: jesper-petersen
submitted at: 2026-10-24 18:41:53 UTC
last updated: 2026-10-24 18:41:53 UTC

Witness: 13 independent points log in to add more points →

(66695433352569486763459/1943089157835225, 877355552541916550486654835553198/85652312475618268084125)
(1097546904603339870323/13277722441, 36360900802757256184737536726790/1529978679153989)
(4373883812783919231329491/6113958369158025, -9146963370213831025546721781468795586/478061440579322304985125)
(102624099730370899450063/1409720063045121, -31109585327612759057096870549598500/52929686410187180849919)
(6322580762612978920342483/141261620631841161, -11791816144677499410753536619666314110/53092879085963933908335909)
(1883971833813392432627/48430506395209, -45880782427488891142312454701530/337037725397057658427)
(3489079533839000037839/101655898827025, 10142077483484314200872561363688/1024941025408032346375)
(330115469076200041644067/9530419408033689, 32546390935194166068985969707902630/930396396744099586455363)
(264927420439655256293150441/2924222437281640000, 4193536955629077294856732521951105328011/5000525054914859082712000000)
(1382513299250611817720574619/2157766817258549025, 51400894194353032012505462315654704772278/3169616836955021813927192625)
(61471381701815036935193/967095928563556, 13992884055487521566238152367709155/30074909724393606038296)
(68082770321656705670321/379990694222500, 17702217082492036454336542259455881/7407291599222170375000)
(6331920499666364884597775963/98247230777134240321, -464344767861225281168869637184596314501890/973824007940397000833404329631)

Commentary

The rationally 3-isogenous partner of the Burkhardt-quartic Mordell curve constructed using parameters (u,v,w)=(13/9,4/9,16/9) and specialization t=119/4. PARI/GP 2.17.4 returned matching computational rank bounds [13,13].

last edited by jesper-petersen at 2026-10-24 18:41:53 UTC · history