curve #6491

y² = x³ − 356791993462936282202847x + 82483064201637095874073487969100930

a-invariants: [0, 0, 0, -356791993462936282202847, 82483064201637095874073487969100930]
c-invariants: [17126015686220941545736656, -71265367470214450835199493605303203520]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 298924201245331810645987537174751475783039180013627808551556424
discriminant (Δ): -32225186328753991112408277771182149727125855533174090528732494133673728
Faltings height: 12.3663
naive height: 174.3189
primes of bad reduction: 2, 3, 11, 19, 199, 2008121, 854639069945171972401, 58164312337319119233735288847
regulator: 3507692373111666049.05211655466379882229932763024426543841236644464273655817686
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:53:11 UTC
last updated: 2026-11-04 04:53:11 UTC

Witness: 20 independent points log in to add more points →

(745450158733221/25, -20348923391510537211894/125)
(-637000077783, -226461131552584962)
(76642590258709/25, -659092633390875227302/125)
(1263693043258, -1284377022045453446)
(733648488058041/625, -17685206865672831825714/15625)
(12112781780865/16, -31759492349514544065/64)
(59900990254485/169, 51490096232399835750/2197)
(194662397318457/256, 2051470070093291470227/4096)
(1099770464765013/1369, 28397098809496139789286/50653)
(-4135787568354399/6241, -82367411936321749691718/493039)
(84497201632714785/299209, 10664919448007183922002310/163667323)
(54795862870663/289, -722892703935253974056/4913)
(-10021817220718719/16129, -519354666496500416635482/2048383)
(-405209548263, 400656080541235662)
(-2633064700887/4, 1433562953691497571/8)
(1042752383635398465/3389281, 267197925281990796030107610/6239666321)
(92006595532323/289, -167434103348891315856/4913)
(5208342249/4, -2291109672881692611/8)
(-123195576367883/361, -2781908639837378611642/6859)
(-2799043108184754531/8508889, -10059316129093280274516582126/24820429213)

Commentary

Mestre ten-root construction, roots 0,5,15,24,25,32,52,128,148,195: 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 >= 20: the 20 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 04:53:11 UTC · history