curve #727

y² = x³ − x² − 54464655817391902854967509221724105157905x + 5530967500672460442450920841909843725925195007329680094248661

a-invariants

[0, -1, 0, -54464655817391902854967509221724105157905, 5530967500672460442450920841909843725925195007329680094248661]

rank (lower bound)

≥ 26

torsion subgroup

trivial

conductor (N)

4888582659985617998857424514806026149612847960176726261100340026980982135342391250413165380935195576

naive height

293.2541 ★ record for rank ≥ 26

Faltings height

22.4174 ★ record for rank ≥ 26

discriminant (Δ)

-2875483139480321576368269142769509735438176888471736059298354220377101197933756025778727197619325375774216213809062116496128

★ record for rank ≥ 26

primes of bad reduction

2, 3, 7, 17, 41, 47, 59, 97, 101, 103, 1908037347922842439489, 7819407193525995498167944261753314952797921630748710245075403853

regulator

58866807955894177051400326544.6348902360842568606251120101539725773708620967930464731225338

submitted by

David Renshaw

submitted at

2026-09-13 18:33:36 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-13 18:42:10 UTC

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

(292957795934780892413, -3836406001880235181410118775018)
(100985156217570653631, -1029899280076620692674825537494)
(-153172483112701681635, 3206204990634440322101499231894)
(-54583371874928404857, -2888115197704543095392722651998)
(-271528415620049653995, 548186949410770224904232228394)
(-273196107389114360359, 142140397950314649470051913586)
(256359302507642489345, 2901103801194164183248918546994)
(153059323807210924967, 883396206949595477996357756350)
(120009974405815168989, 850350798290853174340340298858)
(-32779331663462378987, 2698344238804775071849975580118)
(-62951895723674707833, -2951295677273665408298264719710)
(178295194608075775917, 1219851830650830866187176583210)
(-2179827400414414198145/9, -57366465617750334136031849351338/27)
(32563925662168574061, 1947284156345792351081476797654)
(226444750555163054559, -2192986670397717623263849106358)
(-6500913795978521639, 2425853003013381542269672400754)
(77152109092030186413, 1337216641392152076369180733482)
(70504101636397845333, -1428793077518496484746606837462)
(-200555908343741529347, -2896080355802156284670138529558)
(-589851620581178383515/4, 25744414572423573643480051551027/8)
(1233018645480054233551/9, -21611243793186661538367852452666/27)
(492219594235573962741/4, 6655810560705629646567116371611/8)
(68926966549996077573, -1450638120948239480966854528278)
(-77611574881492319611, 3048041584915132225627606638358)
(222349671465669980149, 2100856557071513707482401347242)
(195664170794379194615, -1537883095613815301571228793106)

Commentary

Specialization T=-311/519 of the family identified by wgxli in curve#302 commentary. Points found using 17 reconstructed generic sections and minimized quartic searches.

last edited by David Renshaw at 2026-09-13 21:20:54 UTC · history