curve #6167

y² + xy + y = x³ − x² − 20967825830676242381168x + 1049906339210514909937857261778131

a-invariants: [1, -1, 1, -20967825830676242381168, 1049906339210514909937857261778131]
c-invariants: [1006455639872459634296049, -907119077073355831806882605821973193]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 19790611384047910958418807384150866111612560217535236970
discriminant (Δ): 113788885721116751303535254279442348583924098910605730046279929036800
Faltings height: 11.7917
naive height: 165.8054
primes of bad reduction: 2, 3, 5, 7, 29, 5494134725747, 41438540727817, 1585972493424107581091963
regulator: 729226925067455932207458.8192822595874263602839180195139509331520834643669569
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:39:09 UTC
last updated: 2026-10-30 14:39:09 UTC

Witness: 19 independent points log in to add more points →

(-140131485370269694147/2347305601, 5197725320303107975530919499409/113724609062849)
(-45664090007519431987/325477681, 205942819228572118369061730201/5871942842921)
(-22047857600262907/12866569, 1520811255726888563496581163/46152383003)
(-433097155402891/2809, 3699660033828541214589/148877)
(41347617187173/361, 84251356854542136539/6859)
(173760483563093/3481, 2320288704401108058195/205379)
(535723345209605/841, 12100900775974373009013/24389)
(1369357183846421/24025, 23541423931284047336691/3723875)
(66939723957464309/1113025, 2969427094463251585870263/1174241375)
(359377060396551365/6538249, 133184046083019469983677781/16718302693)
(709757678181763925/2343961, 537866576691451061787147027/3588604291)
(812225328739909385/102839881, 31021575425998632778292309037/1042899233221)
(5601515185716686867/107474689, 11066676113593233598851770733/1114190100863)
(5958294093388343525/52519009, 4361432270073030286779554871/380605258223)
(16098699469336757525/20620681, 63544774599369039862860694437/93638512421)
(206503443823409936429/3400405969, 142972947303542357070817123017/198287873270297)
(1333211851953757651421/10322356801, 23364077853643884344653006912623/1048741128624799)
(7544264507825807360445/47429435089, 430764929056768961414580552858599/10329324661987687)
(5735832129723243720243085/26329305801681, 11157815443576468271724982057893901193/135101170893337762329)

Commentary

Mestre ten-root construction, roots 0,9,30,34,43,48,71,84,96,121: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 19, proved: 19 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-30 14:39:09 UTC · history