curve #6073

y² + xy + y = x³ − 1533116873568590938x + 746372187527267567939224472

a-invariants: [1, 0, 1, -1533116873568590938, 746372187527267567939224472]
c-invariants: [73589609931292365001, -644865570133943593596428491525]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 1027456936062822754905350724274248877209558
discriminant (Δ): -10030191822290724909508782535593498730100636153928498108
Faltings height: 9.3380
naive height: 137.2777
primes of bad reduction: 2, 3, 7, 11, 17, 61, 4726157, 64824132232123650171832733443
regulator: 16456381910784143844.79213298261212750089522371632206090135644840239513
submitted by: Michael Rubinstein
submitted at: 2026-10-28 21:14:49 UTC
last updated: 2026-10-28 21:14:49 UTC

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

(81041959408, 23068202032447895)
(352350934171, 209151073135865873)
(-631842771513286657994/1268342206849, 53188240969226807520626084686724/1428415871748791743)
(-140677916659964/249001, 4702266420558521514617/124251499)
(-135550367748479/1324801, 45800464598213821711705/1524845951)
(6662264431/676, 475288112932935505/17576)
(104126726929/64, 25898598538946971/512)
(295029437299/625, 176672398557916793/15625)
(1313700671383/3721, 3582453922356021635/226981)
(1558586750491/1521, 940417785061664645/59319)
(1955265222793/9, 2734015736839582397/27)
(42659363395819/52441, 73563759618556830797/12008989)
(1126656107994433/2399401, 42378304595205757824479/3716672149)
(8302685671910320/23068809, 1720798425210022581267752/110799489627)
(32742610595314855/29041321, 3323573181057724871394467/156503678869)
(1691788583096874271/534811876, 2052895325547573766525493023/12368059444376)

Commentary

Mestre ten-root construction, roots 0,4,26,38,43,44,51,63,83,96: 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 21:14:49 UTC · history