curve #1137

y² = x³ − 370268515769641529187x + 146347819865452839744523629536866

a-invariants: [0, 0, 0, -370268515769641529187, 146347819865452839744523629536866]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ
conductor (N): 81988082219777126188462072560
discriminant (Δ): -9249190796881710214856831497600002318742226857585765207325958144000
Faltings height: 11.5286
naive height: 161.6502
primes of bad reduction: 2, 3, 5, 7, 11, 31, 41, 89, 107, 109, 127, 131, 137, 163, 337, 1279
regulator: 35628697642.57981255965173108151241631110098387752269159035
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:14 UTC
last updated: 2026-09-23 21:37:14 UTC

Witness: 10 independent points log in to add more points →

(850458130097, 784187466256529280)
(146152966257, 56693563271115520)
(32251357922, 12959645450812830)
(38151629762, 13702297515202890)
(12677776497, 11987130052613120)
(-333477283383/16, 770691585902048635/64)
(994545732973290393/243049, 991819858363919293950693760/119823157)
(-13164907518, 12204126246933670)
(-19757334467647/529, 126489491575017056640/12167)
(-42557395279, 9221086030423680)

Commentary

Obtained by computing a rational isogeny from curve#1085 and mapping all 10 independent witness points to the global minimal model. Isogenies preserve the Mordell-Weil rank; the image points are independently certified by the site verifier. Torsion computed exactly with PARI/GP elltors. Conductor is unchanged by the isogeny.

last edited by David Renshaw at 2026-09-23 21:37:14 UTC · history