curve #337

y² + xy = x³ − 162268279768512229548027479226276x + 417944668872887979025847241579270199663848529680

a-invariants: [1, 0, 0, -162268279768512229548027479226276, 417944668872887979025847241579270199663848529680]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 42591546162844993850851508738322575321783987912035384306194523783699590
naive height: 234.1140
Faltings height: 17.5584
discriminant (Δ): 197991066515791420644080408315411349123160046175556275120584559630465306544722943095166864250265600
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 23, 31, 43, 157, 1019, 44249, 460589, 28217543, 27038992544935731914128602109293389
regulator: 2334351474430798085.826291464877663755778312160887020417773538122218770317579
submitted by: 7fff-zip
submitted at: 2026-08-24 19:18:42 UTC
last updated: 2026-08-24 19:18:42 UTC

Witness: 19 independent points

(3404128207829471448, -6280667815176558207747997284)
(2019237841998067848, -2869277706439965574463152164)
(4003885097932168432/9, -8008392140607804427928828428/27)
(-6837631768653672, 1098996906036260592329676)
(377771325502728, 597242226991339938325596)
(1301581383017930, 457104280931338878158600)
(2375698019031780, -214132057902391665984240)
(15074541312221640096/1225, 22872483220024242041000638764/42875)
(134167592165328562/9, 30925104798017596474014232/27)
(208133112480715498356/12769, 2092767663218217876382096470732/1442897)
(63867824944237878, 15829636965653541388717176)
(28074782029842938558867592/434013889, 145960227219834149035930958532697005852/9041811349537)
(-4906247057172370, 1046887479457944458658680)
(61965565735548780871982898/2772127801, 411782275448622558381105958744685838786/145955300850451)
(-684523190213992, 727117818176270670094796)
(245274189561110718911689320/1158245089, 3834424479051508503075678346500684112860/39418555113937)
(89223780640476173867088/2825761, 24578915930607842272732643125222236/4750104241)
(82864824052009721736, 754319463202526821026476070300)
(134328903237790271574/4489, 1422122018123538358094814731862/300763)

Commentary

New in this search: Mestre quartic family with roots [0,44,53,78,525,534], T=8551/46. 19 independent rational points certified exactly by Cremona-Brumer quadratic characters; quartic point box [10000000,10000].

last edited by 7fff-zip at 2026-08-24 19:18:42 UTC · history