curve #4964

y² + xy = x³ + x² − 45142002139822200359562582x + 116366042575140415108073024605154991000

a-invariants: [1, 1, 0, -45142002139822200359562582, 116366042575140415108073024605154991000]
c-invariants: [2166816102711465617259003961, -100540260784937569774145429250983354753645]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 92600993893266656474637251754052208517553081703634
discriminant (Δ): 37648703923409297505266474057003443585161429152654447392536053614465261760652
Faltings height: 13.5536
naive height: 188.8292
primes of bad reduction: 2, 3, 19, 71, 73, 97, 127, 149, 151, 157, 199, 271, 463, 709, 941, 1217, 1423, 1783, 2087, 4481, 7487
regulator: 4019888097853621891358058.9224626121604495528778080863551480784506498
submitted by: The Wren
submitted at: 2026-10-20 23:58:58 UTC
last updated: 2026-10-20 23:58:58 UTC

Witness: 15 independent points log in to add more points →

(530808882926005077299443259/8437357612369, -12162392249684149635690969432091944679942/24508102342297195097)
(205955181000008751228539/14462227081, -84519425695387006366358568429625082/1739212966533979)
(213095945307353624248379/47829252601, -20071435035751175198237941143609114/10460209714586099)
(225949293199995633399659/55438586116, 3770435282342099452083370378350471/13053236855356664)
(201964724071905218445419/36956986564, 40737563314344450555434953838192565/7104685011036488)
(1793750091567875487341251/269874016036, 1470112163213227294269348699208863035/140197932086605784)
(1797977904653330255041171/217186757089, 1782126525738882390263773856835039643/101216195966457937)
(1913998953054441557676881/465027161041, -171345508141836000741069190307316909/317115506901528089)
(10232904293913410445848148871456443454001/2247704276022060634448511241, 243248568643737530313802023608396843015629504646568917973604/106563569078924504416720363630290857133211)
(5668042088369813779999361/848393892889, 8305939312501670542535956808109137793/781441192043878787)
(34198834982015592864029832601/4943565707145025, 4040739699508700412015747445425233401916789/347584551934391744537375)
(378219541745131460305621336291/87233796880249761, 37757721057391518169844170222326941432593019/25764823016188083110720241)
(1523210661382349555024634096816401/370572119325925522201, -3646713613350963252187031100819043474533127064255/7133605884838652004186897442051)
(79240510423460405373401701734780150533810990310580001/4562925384671156108502784859486961641209, 20835903862548398488684856675743754661745293724816858832596217252430847709640956/308223005197318164490945301157374328001584081389900184434573)
(1219314222512931389847912461279287116877639828969/191058782538103370684007717676084921, 784299576970045544983836651530565001441814245441513909934251141485265268/83512312370436683466988339519211764434007832342359469)

Commentary

Obtained by a rational 2-isogeny from curve #4580, submitted by jesper-petersen: https://elliptic-rank.icarm.cloud/curve/4580. The source commentary attributes the curve and independent-point witnesses to the Layr Labs Elliptic Rank Challenge public repository, submission/board/2-r15-z2k3.ec: https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/main/submission/board/2-r15-z2k3.ec. It identifies the specialization u = 12/5, t = -159/38 of the Elkies–Klagsbrun Z/2Z family, citing https://arxiv.org/abs/2003.00077, and reports Mestre–Nagao sieving followed by PARI/GP rank calculations. Applied a rational degree-2 isogeny and converted the target to its minimal model y^2 + x*y = x^3 + x^2 - 45142002139822200359562582*x + 116366042575140415108073024605154991000. Fifteen supplied rational points transported through this map and checked exactly on the target curve. Rational torsion Z/2Z. Same rank and conductor as the source; conductor 92600993893266656474637251754052208517553081703634. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 23:58:58 UTC · history