curve #6499

y² + xy + y = x³ − 260980960435079291803x + 1640448180938324734668015822962

a-invariants: [1, 0, 1, -260980960435079291803, 1640448180938324734668015822962]
c-invariants: [12527086100883806006521, -1417347228349503199904491380049165]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 51905558242207374520110452789771714456654440002866497134
discriminant (Δ): -24898161988738517824555622215103275533996915089455007078231388
Faltings height: 10.5906
naive height: 152.6682
primes of bad reduction: 2, 3, 7, 47, 3917, 6712945886802542643348784749325054461723937871273
regulator: 288039943129525366.0274387334485100741773686063343377704851087876441787278706
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:54:07 UTC
last updated: 2026-11-04 04:54:07 UTC

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

(1283440331821, 1453881780085306430)
(335695883611, -194278779231382870)
(10163500756795/169, 31337087862250437086/2197)
(338692734091/25, -96121406460100754/125)
(34125359739241/3025, -60440246327042085076/166375)
(2730761019490/289, 660584989623547249/4913)
(58390010243509/7921, 241322019804283848838/704969)
(97400220469/25, -103289969988603932/125)
(76001536181641/23104, -3175304010206754139481/3511808)
(16485858648077083/25877569, 159845026707635488401886198/131639193503)
(113694588715/1156, 49945179393266899379/39304)
(-148061720, 1295795669400038)
(3634730844215200/1261129, 1352651341093963569044282/1416247867)
(-2402231432, 1501174826065046)
(276431742354373/38416, 2767345126035686543495/7529536)
(-1772884233132707/121801, 65238963399534245664404/42508549)
(1074448587688/121, 202597759322984131/1331)
(9077561932, 139235518863506)
(11152392913, 342009792923306)

Commentary

Mestre ten-root construction, roots 0,1,13,45,55,68,89,135,152,180: 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); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 19: the 19 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.

last edited by Michael Rubinstein at 2026-11-04 04:54:07 UTC · history