curve #3178

y² + xy + y = x³ + x² − 678876005230095721881793359044232088047786825892457820384x − 4283665841626318052298369662438708349320976911262254997412844078356382776145465697279

a-invariants: [1, 1, 1, -678876005230095721881793359044232088047786825892457820384, -4283665841626318052298369662438708349320976911262254997412844078356382776145465697279]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 15066602282658023728483787194675827223128239968819300284366123156642
discriminant (Δ): 12096913104633311026557368754884602008026472720324030217675130826410558508676436250286535046822070515507249678527245594370397131857411863336749870994076361004160861699375104
Faltings height: 31.7210
naive height: 404.1937
primes of bad reduction: 2, 3, 7, 11, 13, 31, 41, 43, 47, 71, 79, 83, 103, 109, 113, 151, 211, 229, 401, 863, 5879, 61519, 169991, 1702817, 893629932007
regulator: 181954427427132124.2018690729154970893551188278439040555959079726
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:41:04 UTC
last updated: 2026-09-26 20:41:04 UTC

Witness: 13 independent points log in to add more points →

(446059616549795336909017463365, 297397064504812433163596573335178838956530037)
(-12375836719772420568752222073, -1490802832147565877844033230089121413962233)
(1826959670585843499691417248829/9, 2448347819170965988658492784709565357814094423/27)
(65680589109961745673557899717, 15312395786439360534357639416024257430997365)
(116369842053518295182201621233, 38634040376262857464278035411127739147629641)
(-5304037150262150068604301209971031/758641, -229797015182989218517592150128496662267403389914505/660776311)
(81859978737699074920339817337, 22106376234493431270895170596857175205301317)
(2038570694390088224370464288236054069/45468049, 2282126856567974939581448420041453571269955254035126643/306591054407)
(-1074783225295629767199693215015/64, -790026108343960965832009941959060355936858911/512)
(-7194279452306502868997617367, -477492205565902502106161306809817097689659)
(3471366374694413821628748408829/9, 6452696333648245255627881053225028423216647063/27)
(-347533664927181860863766546611/25, -196342541868617824039422661069511659059238623/125)
(10939967420343528192620066958384464463237/178360094929, 1024087302667911690474786432818003065724749335695951654616435/75326283811079783)

Commentary

Elkies - Klagsbrun (2020). Source: https://dujella.github.io/tors/z2z2old1011121314.html (curve entry 4). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:41:04 UTC · history