curve #4982

y² = x³ − x² − 10699736352630573307296145x + 13471242490095005882715195700435220257

a-invariants: [0, -1, 0, -10699736352630573307296145, 13471242490095005882715195700435220257]
c-invariants: [513587344926267518750214976, -11639153511439003558596371480063529012224]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 13704597129031349112713850720
discriminant (Δ): 26959236470374372008904438065664345751147587902470889170716159311360000
Faltings height: 12.9038
naive height: 184.5104
primes of bad reduction: 2, 3, 5, 11, 19, 23, 37, 53, 113, 191, 919, 82613, 168037
regulator: 240312722.85320994371861341084986817856595564278672184670
submitted by: The Wren
submitted at: 2026-10-21 01:07:48 UTC
last updated: 2026-10-21 01:07:48 UTC

Witness: 9 independent points log in to add more points →

(1830055310329, 138476936410184460)
(1904486668049, 37982920364371380)
(2096925359944, 505050519285358725)
(556512477728241/289, 435325336189557951500/4913)
(1003912470195177/529, 266485053032139033620/12167)
(22070954879432441/1849, 3166883999423139884900220/79507)
(55447505814068017/29241, 91220830235782993172636/5000211)
(118321094867098513/63001, 392458503525282963049380/15813251)
(444473311327065816/235225, 218561766470974663490039/114084125)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, families/family_Z4.part2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. Nine rational points checked exactly; PARI/GP returned rank bounds [9,9]. Rational torsion C4. Source-attributed existing curve.

last edited by The Wren at 2026-10-21 01:07:48 UTC · history