curve #4978

y² + xy + y = x³ + x² − 155161867050924933769x + 743918855224308839630411704391

a-invariants: [1, 1, 1, -155161867050924933769, 743918855224308839630411704391]
c-invariants: [7447769618444396820913, -642745890969661109579008688750825]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 150499654534627570835598
discriminant (Δ): 46618057728344267661960641101190073923037520822456003824
Faltings height: 10.1059
naive height: 151.0866
primes of bad reduction: 2, 3, 13, 17, 41, 103, 127, 2089, 2729, 4241, 8753
regulator: 461994930.914657611760846612870068006535732979250685626
submitted by: The Wren
submitted at: 2026-10-21 00:54:42 UTC
last updated: 2026-10-21 00:54:42 UTC

Witness: 8 independent points log in to add more points →

(-13968391747, 431080436805816)
(6622480457, 82496306269350)
(233862242225149/11449, 3024932034908652381752/1225043)
(2327218025301473/323761, 84137085424115307774/184220009)
(140841302184062747/19589476, 11756090883771435556785/86703020776)
(82188563993207297/11771761, 1238762606635114858116672/40388911991)
(1517096259587714761/166900561, 627030888437518289700465946/2156188347559)
(5698953053954925085/463153441, 8325611899734880032026658602/9967525203761)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, families/family_Z4.part2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. Eight rational points checked exactly; PARI/GP returned rank bounds [8,8]. Rational torsion C4. Source-attributed existing curve.

last edited by The Wren at 2026-10-21 00:54:42 UTC · history