curve #5856

y² = x³ − 844659558888969536291618424567x + 278714006110937227078411211791343757254942970

a-invariants: [0, 0, 0, -844659558888969536291618424567, 278714006110937227078411211791343757254942970]
c-invariants: [40543658826670537741997684379216, -240808901279849764195747286987721006268270726080]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 5819267853110420154890366735286544171416196785322380671321316665885175203177478347976
discriminant (Δ): 5009411984528749202774584177470186392168547511102635598054871668127267140501675041244876032
Faltings height: 16.1490
naive height: 218.3398
primes of bad reduction: 2, 3, 7, 61, 5107, 5153, 1758673073707969, 64916681682324402712972707812433037359687444023609926367
regulator: 1571639839654745583155087.45033233975303515074227519351361646447496406340417375178988
submitted by: apkorte
submitted at: 2026-10-25 13:12:12 UTC
last updated: 2026-10-25 13:12:12 UTC

Witness: 23 independent points log in to add more points →

(400140502141191, -2190655335354102165612)
(-311770413262539, -22621881778523540047542)
(641321014505451, -886959060166350263652)
(-562270685737362, -23997507798719700488964)
(355604786701234, 4828762091069806386014)
(365997995144985, -4312469280454662677910)
(649525552624809, 2027342135532900957414)
(5780767988836813/9, 29236764845375278503326/27)
(83522716404285, -14448128026609993441650)
(1594205000329585, -54624252080729087854930)
(11785465140650313/16, -479766465813352206526779/64)
(14861678231228859/49, -2435779463958109866906084/343)
(287859018827037, 7708692677026688070738)
(-586331039182623, 23924732845870104100062)
(411881037643719, 829889505711096146484)
(-613101302695413, -23793176897777808981012)
(-1052932595101653, -855519462521213670012)
(407459056885533, 1482344831942816443986)
(-1027842507788943, -7811291993171481908562)
(32129627228098393/36, 3316608116390412417853349/216)
(7264060315633341031, -19578037630902093994283960572)
(838319419295170113/1024, -382007375331011799135378591/32768)
(1645267718747989, -57815289847419623060374)

Commentary

Specialization at T = -79/21 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -129077164062048809835900T^8 + 378319182388641891290880T^7 + 136088129918041916509980T^6 - 755961738076060064377560T^5 + 619450622573441931032760T^4 - 1170861959203596215660160T^3 - 1734751811488996657912320T^2 + 1314050376486783211611840T - 219065110811691252584112, A6 = 18245050345346515787515057919752500T^12 - 83505322013254826025359146692690000T^11 - 2407103439528010819936132599838344T^10 + 465220783429849849086641780587731600T^9 - 185239629309258027189063745343474940T^8 - 1503733795718269687786747306505656800T^7 + 626403979186422524394828379594167600T^6 + 2519798602559624814688441138399867488T^5 - 832465009702733039267308016691100320T^4 - 905573011434584900869203229631278080T^3 + 930446348033589270658482617988215040T^2 - 337397396218794978267255003741415680T + 42647307174629128234916120026233216. 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 13:12:12 UTC · history