curve #6484

y² + xy + y = x³ − x² − 76094048236367904720506597x + 257166908565948460850249142173326259485

a-invariants: [1, -1, 1, -76094048236367904720506597, 257166908565948460850249142173326259485]
c-invariants: [3652514315345659426584316641, -222192209000963033860196203370334258770385]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 80043266134234367016571902619913942762718535666698465277063660482
discriminant (Δ): -371349885848318138853423248205817314992901319998717759199621733582683667693568
Faltings height: 13.7142
naive height: 190.4087
primes of bad reduction: 2, 3, 7, 13, 739, 23669, 9148957, 94320931, 447911121627965663, 7227948420235368763549
regulator: 24318006525217584197.443925281143923916306997604557935069655517884153158194673449
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:52:20 UTC
last updated: 2026-11-04 04:52:20 UTC

Witness: 21 independent points log in to add more points →

(484758981684847/81, -2926735447221230093792/729)
(27034819108351, -134012049221092391968)
(2868790678481911/1225, 410813581173239511792736/42875)
(4367085119428855/841, 34920850589635339511200/24389)
(50403606201343/9, 69696793253977085864/27)
(150148277789131/25, 512232639254113176556/125)
(2612618903974943/289, 86233196424036677554976/4913)
(125629053739446295/11881, 32628758731694503179098336/1295029)
(6962901864412207/361, 529740805024376064582968/6859)
(8428299116447495/1849, -178141974229909198256800/79507)
(8030911701280207/1681, -112659545942861037912320/68921)
(-25152543985450697/2809, -2211159160916640466595360/148877)
(5671558582374379/1225, -86880485033486292060128/42875)
(333052872712116623/69169, -28215666729444722588117536/18191447)
(-501244254581301457/49729, 3729234441511289914905040/11089567)
(389530233356458986775/97397161, -3940688461625029024533007787360/961212581909)
(7573883346403, -10738043819179201156)
(79731868546936719487/16008001, -84078326643798811118900745440/64048012001)
(8432802058176751/1681, -89403237059480951792288/68921)
(855718480382945/169, -2854306668476559083900/2197)
(18561847632959558287/3690241, -9181173363333663823533624688/7088952961)

Commentary

Mestre ten-root construction, roots 0,27,35,58,72,80,102,109,152,193: 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 >= 21: the 21 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 04:52:20 UTC · history