curve #6070

y² = x³ + x² − 146896658335109020x + 22015278567031592898700304

a-invariants: [0, 1, 0, -146896658335109020, 22015278567031592898700304]
c-invariants: [7051039600085232976, -19021200724221533864988460480]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 705887769261288306123559077880860758309448
discriminant (Δ): -6509500556737864265490714405231687358057248903383808
Faltings height: 8.7374
naive height: 130.2307
primes of bad reduction: 2, 3, 11, 3583, 85507965496198979, 8727268724481204701
regulator: 676798554777838267.4720025937223963719063566441319079710603303588113357
submitted by: Michael Rubinstein
submitted at: 2026-10-28 21:14:14 UTC
last updated: 2026-10-28 21:14:14 UTC

Witness: 16 independent points log in to add more points →

(356866018, 3878264790390)
(415492643, 5719179269760)
(-345337760332602883/1604322916, 424608007244653165150934541/64259550077464)
(-440966640765664/1442401, 10727948639848035118644/1732323601)
(-4499788072654/17161, 14656628401128980046/2248091)
(-65209754554/1681, 362444215062569574/68921)
(34228626854/289, 12310484038203978/4913)
(43064266118/169, 2334687994463358/2197)
(90746063093/4, 27332547584877657/8)
(130850920490/529, 10977003852911934/12167)
(614790055328/2809, 87924469817062692/148877)
(9856827380162/61009, 23823648279705423930/15069223)
(26845625149853/98596, 45962009003408534685/30959144)
(653112407748923/2968729, 3008883641506111349748/5115120067)
(738014286682358/6713281, 46657106509132346058150/17394111071)
(362159905064505134/1594005625, 38627038687123766329203402/63640674578125)

Commentary

Mestre ten-root construction, roots 0,6,15,35,42,44,55,75,81,109: 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 16, proved: 16 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-28 21:14:14 UTC · history