curve #217

y² + xy = x³ − x² − 5392334889373133836717x + 152408175069865597518128713267441

a-invariants: [1, -1, 0, -5392334889373133836717, 152408175069865597518128713267441]
rank (lower bound): ≥ 16
conductor (N): 3865572244758359427703312649778700630767441782587550
naive height: 161.7314
Faltings height: 11.1105
discriminant (Δ): 237289500362812252947418258536952148541504393969458434158062500
primes of bad reduction: 2, 5, 11, 43, 47, 113233229, 2477679059415479, 12395551355742150820531
regulator: 1035675008217125.210650282138077450989277944286150483848078016626924898
submitted by: Edgar Costa
last updated: 2026-07-01 22:38:38

Witness: 16 independent points

(-11886849988520801543020205/219020050054276, -54739616447895393532106980635723997109/3241349121289548217976)
(168878841911/4, 371469986103239/8)
(93228044365251/4, -900155943328475632151/8)
(7162568606295437/169744, 4001827171142801549277/69934528)
(-25775659536156/361, -90325109045635789993/6859)
(185255057136930686/4355569, -218161975946970898534041/9090072503)
(10529263292731/49, 32387293583565702279/343)
(864197995016095/20449, -66655557619693066292/2924207)
(-42757882743809/529, 94403070231401230891/12167)
(194866459365465611/4566769, -861267473996490181449116/9759185353)
(-8287086919/64, 6335292054507076401/512)
(4898229429/16, 785813137131910117/64)
(363416598, 12265747209548093)
(-576940395, 12470727566797076)
(700086904, 12191530118261073)
(-4958351906, 13379963865786753)

Commentary

Specialization at T = -2068 of the Mestre/Fermigier family y^2 = r(x,T) on the sextuple (a) = (-44, -60, -6, 110, 94, -94) [r = degree-<=4 remainder in p6(x-T)p6(x+T) = g^2-r, g monic degree 6, p6(x)=prod(x-a_i)]. Located by a Mestre-Nagao sieve of this family. Sixteen independent rational points were obtained via 2-descent covers and a direct minimal-model x = n/q^2 sieve (the extra generators are high-height on the minimal model but small on the covers); rank >= 16 follows from the positive-definiteness of their 16x16 Neron-Tate height-pairing matrix, verified independently in Sage and Magma. Conductor by Tate's algorithm. Proven lower bound on the rank (no exact-rank/Selmer/BSD claim).

last edited by Edgar Costa at 2026-06-25 21:53:06 · history