curve #4228

y² + xy = x³ − 79754469950701904591992983032602101566688375x + 274167332117274700838073990858999307269971448582419348144424515625

a-invariants: [1, 0, 0, -79754469950701904591992983032602101566688375, 274167332117274700838073990858999307269971448582419348144424515625]
c-invariants: [3828214557633691420415663185564900875201042001, -236880574949325341524101670424011852018385955069988664148095583063001]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 445667523771971004518111248205490995553686570820602547907604057815518927174609351664843296976410
discriminant (Δ): -5239926344304762186443428115709005975086173540106654333605808735583772712823958674617131999977910168073777917732348913198080000000
Faltings height: 23.9287
naive height: 314.8763
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 29, 41, 47, 71, 38196596281056749, 563035424011806987299136618729228511340687494179367709500649345243
regulator: 295656944766532255760679154.795617733548791943236618737138028913495370028514326175428955664
submitted by: apkorte
submitted at: 2026-10-14 19:06:09 UTC
last updated: 2026-10-14 19:06:09 UTC

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

(5164056838845195992250, -4807731343779708170848175215875)
(1490115136642206871830, 398287214371865304743010447294645)
(3164586833209432645410, -231234705470648262357816364253775)
(4759325340314557303500, 48930422733741808967880096387375)
(5084378491576216827390, 10060022897782300611406739269725)
(291625301548199881847750/49, -34834018823598973356667951977770125/343)
(13311788895120363123410, 1253551444700482028832784931632865)
(5354968542014039719650, -25338943689903513971287914053775)
(5096134634335082769750, -8799409981140587210550227140875)
(2226197240089021929750, 328102244250521982207734696659125)
(26608757093442770581500, -4122097844735203741659983386870875)
(4829092131779856525750, -40505666835243091224532100680875)
(5131266305028690374790, -5621844235241745776693290093275)
(5314087987232355300750, 20307993480414600768922498630125)
(5266643393844334548750, 14583307266946348610112279862125)
(18198250041510009969302310/3721, -7502858640138222500520522342208826775/226981)
(5585125370332388040150, -54304025870110199797840418899275)
(38690146228588798129890, -7423260457164135848430982863849615)
(3957955688305656792150, -143198063933612787433253629540875)
(5132988311501315121450, 5507614241196722655978178714425)
(4983548064982514667750, 21846238436695742300872808479125)
(414692772875482621225750/81, -4755251102440597496273828927157875/729)
(5134730089229800077750, -5398222333920525782742965320875)
(3533647270693570912662, 190962377966770662130919230110261)
(-7330761085537567123050, -681815789066346295443387197616075)
(5177435654466304087650, 5404380468876026433450358615725)

Commentary

Specialization at T = -18356/2459 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -10669947921935414766592T^8 + 11909462495223346810112T^7 + 29290850738695015599056T^6 + 749826724411797003953776T^5 - 2800296645908250177432680T^4 + 5291011042236720751525024T^3 - 9053743075112346775820704T^2 - 1373686135035664689321352T - 153062929487822447776267, A6 = 424084459520579325699737587892224T^12 - 702971076900016054991835746922496T^11 - 1428460164091780099364248773693184T^10 - 45531551300070380060674898437602560T^9 + 178145138683306119166957105019549520T^8 - 224307849011465835658368095576866752T^7 + 1890363249575504635685922631864051728T^6 - 8402714629932209203714457518206796848T^5 + 10932033646871752694840038266436614840T^4 - 1045636958253350245306227479236779680T^3 + 3911537622199302231442442603656239344T^2 + 397823992391227625022128206594583384T + 16543056563137220502197778370669126. It is a 2-neighbour of the fibration given in curve#4205 (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 19:06:09 UTC · history