curve #4191

y² = x³ + x² − 1877003524652915304586600449793941048564460x + 975260243412999658493303275196323797591848333388056722094183504

a-invariants: [0, 1, 0, -1877003524652915304586600449793941048564460, 975260243412999658493303275196323797591848333388056722094183504]
c-invariants: [90096169183339934620156821590109170331094096, -842624850308831704938754606784723800727077900976821662911361112000]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 446448313509287809070324734585228960192023626696145072346536211063504119846564087558171963578055559964904
discriminant (Δ): 12339566426385275008377177211149670702400666253739000858492009025720143119315020247784676189974611382540926217882119691105504512
Faltings height: 23.1845
naive height: 303.6284
primes of bad reduction: 2, 3, 7, 11, 13, 23, 73, 109, 4705003832992487, 1961986463684002465830810143029032289574150176463092039285383835010095895952673
regulator: 146091809245625170340272562170.67641142638072714666959125825545591553402743486567672688102439
submitted by: apkorte
submitted at: 2026-10-12 03:51:03 UTC
last updated: 2026-10-12 03:51:03 UTC

Witness: 27 independent points log in to add more points →

(352480991320283187614, 18906227095774470994670977204902)
(1119910689520491129422, -16666609297932691243184535943446)
(112696479282831339056, 27661523020567622799333013929636)
(912213599847787085282, -4702736998518075868768000539774)
(1047624766677366936674, 12595695204746080584386796115458)
(2042757621398211659564, 75267097061274770906070335836548)
(923849029529479362470, 5449289955754514487170854999698)
(4006472271077408677706, 240346145455949167065826135354614)
(-840400955450248297945, -44262203415835049065875025974402)
(879421180501015557566, -2170658607131681902062975261114)
(1105749863716811265932, -15866478761781426560204853833676)
(638939146944424935770, -6067272526982552219289874116198)
(675197484412157153426, 3965989891988692103533270286286)
(124555365558644629963529/256, -54523304724302322896118460688305947/4096)
(5637103043893159626962258, 13383938969881531695933836294432701170)
(1497167464063841800391, -38999847445131056200640354044614)
(531816202777546772519, -11289469481144266645532063060478)
(-1244947138891099316050, 37181918775159734205201549248502)
(1080827065059003140039, -14462099282462897059897864859898)
(913851159580644340307, 4810132902591251417799836941176)
(445418383240884550658, -15085697793147535550725104265890)
(662265882354798697136, 4759420178099439989873217316836)
(707657119792047992546, 1168279279782517478200587449214)
(5375338892221430134450/9, 221425534435637492851112832664054/27)
(951211110870677722226, -7105765295076901478125008194286)
(633434365299076062365/4, 208919787548125651634809969655859/8)
(680156204988318993095, -3640520110503213393499169824698)

Commentary

Specialization at T = 57654/12097 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -4437370829345071269937785502500T^8 + 24731294533954602783634178920800T^7 - 24905343694917325040325875752236T^6 - 73371957550294374341624859203376T^5 + 11388134363003403434011617130260T^4 - 31974960227112637988116696606080T^3 - 53087557802286047432098781752884T^2 + 33902036804480905040632762295376T - 4900264269756058995659292474432, A6 = 3583088599917473486225042501788936057800562500T^12 - 29952862178447015859986519449513030145545410000T^11 + 74261855654700672409867052055125389734923941200T^10 + 34915854636953526118546131005709436129082490240T^9 - 320350491450370659682737733420310606418445799256T^8 + 104052954099599046113301816294931102834227932832T^7 + 846552244341643086726187489356042742190493998112T^6 + 378044570078449717959306675143150641014761409024T^5 - 387170892915283488485440730375460569591982158796T^4 - 122143831120324014405241873122119111177801883600T^3 + 159312688808409642522129729739582813691214920688T^2 - 45075775528724488962464647393974897773506063488T + 4205873318402097235409142015354269085411308544. 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 2026-10-12 03:51:03 UTC · history