curve #4224

y² + xy = x³ − 1494270410164589646550689628792966920x + 670871929641747445961232837555705146649381177854222400

a-invariants: [1, 0, 0, -1494270410164589646550689628792966920, 670871929641747445961232837555705146649381177854222400]
c-invariants: [71724979687900303034433102182062412161, -579633347210469793418092641179979701256714990939141771841]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 1328015877675620242749067797948989434490391236328754978415175391077627125050756249555470
discriminant (Δ): 19104388610819094744447973669190338515360068700909657468335218120407525920135618101876177483003332441600000000
Faltings height: 19.7267
naive height: 261.4977
primes of bad reduction: 2, 3, 5, 7, 13, 19, 31, 53, 67, 232581224851787196031785080513655419432376166086234580579404532365359209088201
regulator: 2329259452135505285434879.4983795804048994361071904624381311560877669236424111280792975
submitted by: apkorte
submitted at: 2026-10-14 13:42:04 UTC
last updated: 2026-10-14 13:42:04 UTC

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

(1267015010591339640, -900878651145691673543273520)
(16688542625836457760, 67997185304342941529932648680)
(127902420059191740, 694149470414175740829208380)
(222799888263979020240, 3325568742213022781725639565880)
(888591416928085800, 211431187924536862306190520)
(541799005995831240, -142550784195979466329314120)
(897898411362337776, 230377857774505116253314552)
(-242040039575225760, 1009141058178345884844179880)
(487063929302038970640/361, 7229125277638777775309600266920/6859)
(1034367275607791040, 481594500801314800851097080)
(2683835840429292600, -3999012057188668779263374320)
(550751298937878240, -122304112356752762690364120)
(14488672876910714539440/11881, -1053817765695704466748014070929480/1295029)
(-1098576195623541360, -993279192338452343360672520)
(319635338092863120, -475296124023075436629237000)
(1390441625610348240, 1131969857265142966553645880)
(934689923286175440, 301296944115390244346669880)
(1806646572917407440, -1966747811864034781210117320)
(-51403930834645477980/49, -357104488823679315414605637960/343)
(826067610485828640, 14209744854954133322363880)
(862978154402529990, -155035632993137098000534620)
(2322942419139758952660/2809, -3937856709391309965111343585740/148877)
(174771319116900240, 644247399074367803588165880)
(-675456903728248680, 1171330596755438589059023800)

Commentary

Specialization at T = 307/19 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -48788204994959095721152T^8 + 292967918358775673142176T^7 - 257383034328976566251548T^6 - 697546673797396881668416T^5 + 120168749817393327515900T^4 - 1757567929634520292525024T^3 - 413537200758726229861444T^2 + 1707114327506779370979968T - 562162388405818919328652, A6 = 4093740998135849949800603694210304T^12 - 39254200079660475614530561864102528T^11 + 106574491886043297526892628627228560T^10 + 93605566178607331448553679365937280T^9 - 784525015208748993169852391016411996T^8 + 155163719535379155266150397777650688T^7 + 2578710394897183836702469318351911360T^6 - 11129400918516684422195231027746944T^5 - 2571149805127224053198426441759406936T^4 + 885351238544731577742776581975479680T^3 + 1010399979603017167929810575585795728T^2 - 806575307806361814743442724032893696T + 165741056816888627629082643287877460. It is a 2-neighbour of the fibration given in curve#4191 (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 13:42:04 UTC · history