curve #4215

y² + xy = x³ − 1345076432669331440295292748692199770370x + 16538410542447794548351042856950811912819281638729953352900

a-invariants: [1, 0, 0, -1345076432669331440295292748692199770370, 16538410542447794548351042856950811912819281638729953352900]
c-invariants: [64563668768127909134174051937225588977761, -14289186708674894489872146531557693356377120413768518080372241]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 145159370006636113651476543950798325070663960016825995909894028793437575072639958539379467830
discriminant (Δ): 37587203028827978934621409770806440165557509777604684996745791608004856566183788027112976340378386506108095053824000000
Faltings height: 21.4787
naive height: 281.9054
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 31, 41, 191, 9463, 272045818228399, 1689549936504702447779, 269289845010499341380794935162012655492511
regulator: 120199129353076184786706481.4146464666783363046708490151304595531311902876665884507251156
submitted by: apkorte
submitted at: 2026-10-14 04:09:39 UTC
last updated: 2026-10-14 04:09:39 UTC

Witness: 25 independent points log in to add more points →

(31022102829197834140, -68308703971778626160786294570)
(27254107099871058340, -11114505313443161507032290770)
(130605568740969728140, -1438300912098667340614665620570)
(1082941743889928335276, 35617323216777934161209878161742)
(58021259030717860, -128297964557766239464892315090)
(747682451106726169795/4, -20073455761726752985814383181605/8)
(32211792971574143140, 81449559052721187190686384430)
(-7704446875028354588, -162616592133163642817927280338)
(-10139947296975550940, -170689308953475762628103970770)
(-13637045145696794172710/361, 795271869903149034610410942458620/6859)
(3101418332911007140, -111340009630837989618810863570)
(45396421470982098940, -221430106096393160118811408370)
(553221038200018114435/4, 12588395825160296840827073268815/8)
(5159851082359501711060, -370633616156087213518472553000770)
(2101831666542008740, -117134843478280372887383049170)
(-2169848214851296298748860/175561, 13008881676193256049199642803123625370/73560059)
(34083123980688505600, 101424765412642883633658845170)
(14456812184419951300, -10693854414303986395353522770)
(28837090862505308260, 41600529859801931766300221230)
(-29486747983714112060, -174821473189313904124584652370)
(-5134582430554490960, -152674321856149091526304523570)
(-26021305723258342620, -184173321059307196694626320210)
(-36435628503899962460, -131060345099252869970672533970)
(52968637037810831192260/4489, 14454982442327078126470423988704090/300763)
(27289337547927589864390/841, 2048997122207701765655443771878920/24389)

Commentary

Specialization at T = -2785/864 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 04:09:40 UTC · history