curve #2726

y² + xy = x³ − 1787601681452114389804000049510749791592195487785x + 919831301455888387283167876795422252219907558423929474955754599589003225

a-invariants: [1, 0, 0, -1787601681452114389804000049510749791592195487785, 919831301455888387283167876795422252219907558423929474955754599589003225]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 4528475143582181274801482317230
discriminant (Δ): 77539873250657166354785047006922667669580127524950792443083747499723053879460361556469655943658200981030689606649614044565103284602880000000000
Faltings height: 26.4415
naive height: 344.9285
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 29, 41, 101, 113, 157, 239, 521, 36979, 618679
regulator: 12902220.62075227260129714779280916488850934968
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:19:33 UTC
last updated: 2026-09-26 20:19:33 UTC

Witness: 4 independent points log in to add more points →

(7189397741544483261909432783684507303183499124329633841843331544330/7040557219527856329385297437463037163509449, 7454119555772028929051798437552467417011916829642154678089412961873681479239081386340591541125336655/18681448568883872952806848466259839790198346286952842372892845957)
(14009386179844710063020841062599413350170137669830241980963417256170/17868469872397268895644371860440663555503129, -1178601056247288989115774724437027037516271750325730992041363918819515231519319401178174457268762295/75532010762968692773989001281939245233768140146652103480837396867)
(70755424644626627378597549013825658193099790774526892885222805691024970/90865975315651813029485527397482707099739089841, -79399238715994721822653616788404404357323347223836363153085028528392602239760547009496339342688010404565/27390624785745140865946740790207367254541518722385580140521978231270889)
(-58979920822806427940167983323678174005/108932931164836, 1494973251030490115916446594590896230344519390751593944485/1136943242262922826584)

Commentary

2-isogenous image of D-z10-022, rational kernel x=-382712265091480194077830. Source: https://dujella.github.io/tors/z10.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:19:33 UTC · history