curve #4278

y² = x³ − 52054956130733279043956568788891292x + 4583686313913663671452802129221594160700896899106100

a-invariants: [0, 0, 0, -52054956130733279043956568788891292, 4583686313913663671452802129221594160700896899106100]
c-invariants: [2498637894275197394109915301866782016, -3960304975221405412135221039647457354845574920827670400]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 80434146883505932726005348229594318875747348526136885446814145870458512275918533423530344
discriminant (Δ): -48924228379966546147736842866492934609598432604952731444588226184276407272649297064087580533366310298368
Faltings height: 18.7636
naive height: 251.4318
primes of bad reduction: 2, 3, 7, 11, 17, 29, 37, 43, 745515769574693139059, 95061091737541505740475025201417024601534871192611368373
regulator: 27965673808001077022803098.575100919616114389141500451445836157384698360718022921866668
submitted by: apkorte
submitted at: 2026-10-16 13:48:47 UTC
last updated: 2026-10-16 13:48:47 UTC

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

(108938266123823250, -14343622810030489487314110)
(239079416445688086, -76183661910242907782707638)
(116812366477241274, 9845989353724151003479554)
(129153229687557457, -3868899608219416024496093)
(377496619863645396, -196793751536174273099328402)
(-79855254736513868, 90726634346464274664383218)
(135072178896437050, -4102859793121165066933250)
(106940953224495924, -15488567545227907828842354)
(1263209450861613666, 1398044461224726642370304382)
(-124849674725890964, -95585792255931469340071162)
(699730032540917097/4, -230583101294659922856362667/8)
(167823911302044972, 23965763358342267645847482)
(-89466387771627566, 92329547022009184889655874)
(122164792636883308, -6900699754250948600199674)
(-14113234336335900, -72907742979659594054103330)
(1244354954459611218, -1366232781194934157214865474)
(133560558715849137, -3702298054389063143877357)
(-71312406887700942, 89068480271021843995962174)
(50217070474741382556/361, -40167104733230053776221529582/6859)
(90210657457202096313, -856811692206921972559493918001)
(324217663745304366, 147605448815660059062271818)
(222699139789077681, -63528225680441404229953683)
(548767240503149609500/3481, -3529293816530694055381480132190/205379)
(107238552095394372, 15318061843463908746774018)

Commentary

Specialization at T = 32/21 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -2350796244229826784432T^8 + 40408811499609701188800T^7 - 807647185297995630241792T^6 + 1124780763822387162339456T^5 - 1304766910580579021664360T^4 + 1443539650170624433343496T^3 + 1704268062115609657765788T^2 - 1468913167837302046081776T - 1000351384601036369237532, A6 = 39869593619137462773895754588544T^12 - 1348966735043080317092632272134400T^11 + 31027342372605406058463256507914496T^10 - 139468061144328757444897112746057728T^9 + 1081825061951776499965823580935603424T^8 - 1898481303093606040657233497704979232T^7 - 570283484013752985306584054565000912T^6 + 2509799809819326794944663142993968224T^5 + 574280708473637397501657996019459428T^4 - 2029433790678638479785535621658530032T^3 - 641250037642162230715032671759921448T^2 + 863840080098002956108405878211611888T + 384816973441734885044295380206073844. 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 2026-10-16 13:48:47 UTC · history