curve #4223

y² + xy + y = x³ − x² − 21555825869833463413437304571731860387x + 38394233302214512501329423320027818587957829759922532195

a-invariants: [1, -1, 1, -21555825869833463413437304571731860387, 38394233302214512501329423320027818587957829759922532195]
c-invariants: [1034679641752006243844990619443129298561, -33172617573113338796492563360620007162693107125078985973185]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 225844906993455043280757412501037785360724212266388761170205156457263736701803479546704613860962
discriminant (Δ): 4204924587257814326043938015887585060405620797220188369386915901580497262671926319100156972114317655183522660352
Faltings height: 20.2775
naive height: 269.5047
primes of bad reduction: 2, 7, 11, 17, 23, 41, 61, 375153763410287, 363410221733252303361812769117415788340837473984904761318875507501320219
regulator: 261644586440324123616609946.9582890492175211076080094143524120036565483176614791957351500
submitted by: apkorte
submitted at: 2026-10-14 13:26:51 UTC
last updated: 2026-10-14 13:26:51 UTC

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

(1272644010324870043, 3608676189813566941942401714)
(1418509846090341403, 3266705569507858933022757682)
(838039272904286215, 4573638362719196687763766594)
(2419858463279571163, -634182227863182671602035086)
(1470269769785137923, -3143187287645142127525930638)
(-3896388589116350005, 7951713237557132920727940274)
(607480781632668736079/169, -5958428668846297824012100808578/2197)
(-4585809883793799269, -6388048628065673520996661838)
(62890173834756427971/25, 37019602549044033016131551466/125)
(1978708513665175691, 1867819479260609508917817026)
(2915125093602444507, 573501744872502898971183282)
(3609060915840914835, 2758100881165355551428446714)
(5615934486090476331, 9718925533112536483653878306)
(-14032560210618746139/25, -886678546923591476746162066956/125)
(-112223708495899349, 6388418791712969944577831602)
(201876958245704807947/81, -283930262723331041190483057182/729)
(2443797257736757083, -557563286688102607612444686)
(2829525246665022507, -235078601700900287386541790)
(2810688294137588123, -108974179325904914743038286)
(6532304768138264340099/2809, -136642445803854236428371967497206/148877)
(-5160221943533149353, -3495891023318840572997429310)
(4717954500649975291, 6458500094729073752901350226)
(9813035258844124475, 27781628287717700740795389970)
(188336448004443907, 5860133619050016633367091082)
(2939879932310901275, 657087571856514008376962354)

Commentary

Specialization at T = 961/909 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 13:26:51 UTC · history