curve #6443

y² = x³ − 1259005087182782257x + 544884986523763705214598481

a-invariants: [0, 0, 0, -1259005087182782257, 544884986523763705214598481]
c-invariants: [60432244184773548336, -470780628356531841305413087584]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 513703381698555856869103997914550200450077860
discriminant (Δ): -539613296012976767899939430809365326185278662063750000
Faltings height: 9.1887
naive height: 136.6484
primes of bad reduction: 2, 5, 7, 1289849659093, 2844757792823385553109552181943
regulator: 43295939776222.3464244044368186951203005631071949232524525454529972371397
submitted by: Michael Rubinstein
submitted at: 2026-11-03 14:18:13 UTC
last updated: 2026-11-03 14:18:13 UTC

Witness: 17 independent points log in to add more points →

(350387407, 12114577514095)
(8448245191112, -24555555363206735875)
(-495958512845903/968256, 30952086348586524302825/952763904)
(-56911340303/81, 23986467677792375/729)
(-162996993475/169, 64503230180461177/2197)
(-1176202993907/1089, 912493954676954125/35937)
(-807826491119/900, -832984699400083279/27000)
(14954128654853/26569, -16529405849255896625/4330747)
(25897975556897/27556, 63345113658554956375/4574296)
(4746615613/9, 141799305941335/27)
(-7734951858657/80089, 584734659252232244875/22665187)
(26114871772993/54289, 89951638998467362375/12649337)
(367119678366577/793881, 5550129964225772067205/707347971)
(12180540809/25, 858136976874877/125)
(4837802528737/15376, 25573425466412476625/1906624)
(-557076423563/3481, 5595472887706365335/205379)
(2072024993/4, 45048842124265/8)

Commentary

Mestre ten-root construction, roots 0,14,123,126,130,132,150,163,166,190: 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 >= 17: the 17 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 2026-11-03 14:18:13 UTC · history