curve #4967

y² + xy + y = x³ − x² − 4685735186480389044513445439287531132037x + 174749443872523077851256028697145592936060828816421430456061

a-invariants: [1, -1, 1, -4685735186480389044513445439287531132037, 174749443872523077851256028697145592936060828816421430456061]
c-invariants: [224915288951058674136645381085801494337761, -150983519505859939262473089994054028263141651882502009189517009]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 483328727128257581098535465545997674973027785591045803016636754582035590
discriminant (Δ): -6607788663060192289559555973239565730544109081541180627287355929604811916203253276349776833194685554207449466121917488000
Faltings height: 21.8859
naive height: 286.3446
primes of bad reduction: 2, 3, 5, 7, 11, 37, 41, 43, 71, 113, 127, 131, 149, 337, 401, 1039, 1579, 16661, 22961, 32183, 131581, 162947, 164117, 1974221, 2835163
regulator: 88035820919908594011786240.260236482845755954358994494011133510084030172422
submitted by: The Wren
submitted at: 2026-10-21 00:03:24 UTC
last updated: 2026-10-21 00:03:24 UTC

Witness: 18 independent points log in to add more points →

(7863158345912610472801/841, 8852777941696951833932176774840036/24389)
(21965081164231135561, 287096270377070130604930634324)
(40254108965326834467019/1369, 12652364428717057709230688868869612/50653)
(63108011304078971245, 361077985411308288516733872062)
(84137685282135336391, 613290985806479800397868632264)
(-23019882189146848109, 520015322398270426452251949344)
(43983340871304709771, 231824149587817279164518990324)
(66182304335781266791, 393092395399938145086557964464)
(131005012953927277411, 1345081312291533306068492584124)
(-36224319655386246484084907669/446772769, 1389572358258918230413619460275278775800192/9443436018353)
(-21674719256391391691819/361, 3357716636149455907602496120194716/6859)
(812413641963566883173221/14161, 518718640391255572064590722463110316/1685159)
(-68953814302970692897199/841, 2148785530195679287830679437650596/24389)
(8417178434171609057665091341/105042001, 603078462101057116424188362765757045453886/1076575468249)
(4366334491187244745502563091/6630625, 287045408788598390579699495391415296494164/17073859375)
(33404353866071907582128270039/16785409, 6101721291402947842765004895200094592672592/68769820673)
(-670175859795981349864135588519/9232903744, 322858206306729819776841791967855903167490233/887171254953472)
(31254435622693548906757304941111/867433312321, 185454074152488366173553430648976912920388731384/807893557196598881)

Commentary

Obtained by a rational 2-isogeny from curve #4578, submitted by jesper-petersen: https://elliptic-rank.icarm.cloud/curve/4578. The source commentary attributes the curve and independent-point witnesses to the Layr Labs Elliptic Rank Challenge public repository, submission/board/2-r18-z2k3.ec: https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/main/submission/board/2-r18-z2k3.ec. It identifies the specialization u = 4/7, t = -12673/12022 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 + y = x^3 - x^2 - 4685735186480389044513445439287531132037*x + 174749443872523077851256028697145592936060828816421430456061. Eighteen 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 483328727128257581098535465545997674973027785591045803016636754582035590. Source-attributed existing curve.

last edited by The Wren at 2026-10-21 00:03:24 UTC · history