curve #6498

y² + xy = x³ − 48805101070706591709208900x + 132601598038184469079937867258404010000

a-invariants: [1, 0, 0, -48805101070706591709208900, 132601598038184469079937867258404010000]
c-invariants: [2342644851393916402042027201, -114567780704994895252343408185864127680801]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 7225366625201708988410703605170496823734687573654248410
discriminant (Δ): -155889355157290562131122737946303661724703221986701909423850322980649356800000
Faltings height: 13.6234
naive height: 189.0840
primes of bad reduction: 2, 3, 5, 7, 11, 31, 67, 149, 449, 23189, 184411229, 1005439531, 5235443337652625965613
regulator: 25682404972292939.33811618219981944748013213300988900516698180816468098125491
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:54:00 UTC
last updated: 2026-11-04 04:54:00 UTC

Witness: 19 independent points log in to add more points →

(-312958118671210/169, 32336204006067627732650/2197)
(83795781769532, -764483233120287102184)
(-21876999412225/4, 122879837730714792025/8)
(2366748420658960/441, -46536877260070251061300/9261)
(4488435974906, -1991814452424026338)
(685555467447278/169, -2575480785680052089806/2197)
(38534308730713490/6889, 3364319916397017551585650/571787)
(2831053621404410/361, 104510433786803041068410/6859)
(156948811739235482/14161, 52024941755199124859347474/1685159)
(2263193364730, 5808474881198401000)
(137776938659180/49, 1439078403093216624500/343)
(11975131350115/4, -29199692323842709525/8)
(45335756621000/169, 24019600783852177774700/2197)
(532403118815788604/133225, 57214078086142392123331228/48627125)
(-72956070547465150/10201, 11088942529744260106414600/1030301)
(800062944126800/18769, -29376655192579163522110700/2571353)
(44678496787073/64, 5091012804828694543699/512)
(364157007744620/169, 13443469016682631217000/2197)
(-4156531450936540/529, 67727096325179940838900/12167)

Commentary

Mestre ten-root construction, roots 0,13,20,42,55,65,83,95,119,144: 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 >= 19: the 19 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 04:54:00 UTC · history