curve #331

y² + xy = x³ + x² − 19765899815999953792466328892x + 930706389260018051001267482454403054915180

a-invariants: [1, 1, 0, -19765899815999953792466328892, 930706389260018051001267482454403054915180]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 3793516941765244041911805984731101725684082461834129521696000864322
naive height: 207.0749
Faltings height: 15.2432
discriminant (Δ): 120026114450375268731550080313050637490976047007792951958607127362398246018745515426132
primes of bad reduction: 2, 3, 7, 11, 17, 19, 23, 41, 12348031756441, 62795042146961, 34766641820972554344399273084379
regulator: 1667257467816680568.761756590183763270982336299364955494250659404531856245151
submitted by: 7fff-zip
submitted at: 2026-08-24 19:17:47 UTC
last updated: 2026-08-24 19:17:47 UTC

Witness: 19 independent points

(8948871752189262/169, -396291749135598048872560/2197)
(331631657273881498561/6538249, -4048336062850691996481244470539/16718302693)
(734498777347765/16, -22176824792338565475755/64)
(729780904227973/16, -22541019394000164821987/64)
(24541525333748266947/605284, -208450745296992975096570371071/470910952)
(387922489046025282919/2647129, 4680871983970916113209065564635/4306878883)
(272857891108108, 3981467493175044681502)
(128187235733585194/225, 44594835273094520808275792/3375)
(2825658530774346123631/2356225, -149208357924334609825389518668201/3616805375)
(21723276711930582382/26569, -99827762782453171410618564176/4330747)
(-9462795346882829837/203401, 121337901129172690209953839300/91733851)
(-16080994413244107/6889, 565124956676911235549761142/571787)
(1953541393275260017/38809, 1923397946333882434451317325/7645373)
(117390910883342715/2116, 6804564761696889034824215/97336)
(29165334153265143781563223/521672508361, -7469235222751949106574169084939264627/376787880941391109)
(67188964108724004991/1216609, -116559474111350125762347666616/1341919727)
(107294547606315080210/677329, 742589868815121423327651929090/557441767)
(89023261784394761389/4669921, 7557559331086612408945520058817/10091699281)
(417649578592662, 8094849481736481648416)

Commentary

New in this search: Mestre quartic family with roots [0,115,206,234,731,784], T=9636/23. 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 19:17:47 UTC · history