curve #4203

y² = x³ − 1007654472649017081172739855731473193974172x + 318598214768957788351345970343016621772241932529286310784731764

a-invariants: [0, 0, 0, -1007654472649017081172739855731473193974172, 318598214768957788351345970343016621772241932529286310784731764]
c-invariants: [48367414687152819896291513075110713310760256, -275268857560379529135562918376366361211217029705303372518008244096]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 19834100158182773487723967518986340699072465094418945166043502766109655929803468704702942705080
discriminant (Δ): 21630853614093693837362284794469412620847618653623339834332889415027341016292666403945320171786699383194129255593301231700000000
Faltings height: 23.1507
naive height: 301.7622
primes of bad reduction: 2, 5, 7, 13, 19, 29, 31, 37, 43, 47, 61, 89, 101, 167, 2719, 137293394641841021422073434861271, 124799616451944482251749586040176426183
regulator: 656374557138509808513580005.464499161071905166455463291053535349698695168683795729192627003
submitted by: apkorte
submitted at: 2026-10-13 02:28:26 UTC
last updated: 2026-10-13 02:28:26 UTC

Witness: 26 independent points log in to add more points →

(-295823494465667405407, 24306338937594110714389707632725)
(-943260995230689320082, -20732127399757957818665340059750)
(1790007353138265706548, -65194317084895185097767162916610)
(331112983818642854918, 4610051540908391539737709459750)
(342764851068740513418, 3671559382387656463037421592750)
(2062576472970741926798, 83754948569947799858488817379610)
(349540338516828599418, -3014734723346361398860262416750)
(1320176903400274265358, 35905555096066283352124210826470)
(1173038833975385777958, 27399003774013610296202865040730)
(-457286520686828660452, 26148831682693575154267746615110)
(1454091463747874764497/4, -4608933417012919777469285696825/8)
(1707146948995522586865, 59779669380831674377637543296147)
(-124272549238275770502, 21021483797096343782537225626990)
(-128293320497272993680534492/143641, 1224717694825048273025365694417982870710/54439939)
(4029675727919062917918, -248380970092610885955556703363750)
(-913959677422776882, 17875099148320347424519491813350)
(236017143523517329118, 9691315966409543345230229501350)
(2564809032614132727890, -120855699389748890643658747252178)
(-1077578356549011445207, 12376093641522564941565007714375)
(1346312275570060235683802/289, 1527956649492110878607554494671323250/4913)
(33933838169651761668, -16865458960850622129618440488750)
(250604118075256032918, -9045131309424917461347346276250)
(173567679799392233668, 12203721588679435838613965264750)
(1250878655246798047118, 31865205716383859039025026234650)
(844767590291839652268, 8379606014036269554397140562550)
(-20884576293745025824026/25, 3003620785973389088830409005901618/125)

Commentary

Specialization at T = 388/5 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 2026-10-13 02:28:27 UTC · history