curve #4977

y² = x³ + x² + 31844196315128532495x − 84374832693546045530379142497

a-invariants: [0, 1, 0, 31844196315128532495, -84374832693546045530379142497]
c-invariants: [-1528521423126169559744, 72899855456394911877004596475904]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 130610267722497718589280
discriminant (Δ): -5142125205514529694553834533603834181870468816339928701440000
Faltings height: 10.3349
naive height: 146.7333
primes of bad reduction: 2, 3, 5, 7, 11, 17, 29, 181, 293, 677, 1021, 195541
regulator: 531889666.770741359406843389160628118066589373994782708
submitted by: The Wren
submitted at: 2026-10-21 00:53:20 UTC
last updated: 2026-10-21 00:53:20 UTC

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

(4389533721, 374144229149460)
(3634839259701, 6929921870246067000)
(25366937473/9, 4499430081017300/27)
(779547416169/49, 723465534011914980/343)
(12805876313641/576, 47108526270746227595/13824)
(96933832408409/12769, 1112849298702091430940/1442897)
(2120449382833269/770884, 104921078327459350529265/676836152)
(4813930359123021/116281, 336894281705000389513080/39651821)

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 00:53:20 UTC · history