curve #222

y² + xy + y = x³ − x² − 3331019292820252299857x + 88471823270026370222046861281089

a-invariants: [1, -1, 1, -3331019292820252299857, 88471823270026370222046861281089]
rank (lower bound): ≥ 18
conductor (N): 66208178266486068165130805597508803777331470494813390770
naive height: 160.6436
Faltings height: 11.3835
discriminant (Δ): -1015940660555052781009392764803757490793791575413533348544972800000
primes of bad reduction: 2, 3, 5, 37, 1601, 20646107, 548313117529, 1097006657722059559605800613523
regulator: 13449739871232258.441561791518281871515000729231548101061803269736297924462
submitted by: RoyManami
last updated: 2026-07-01 22:31:41

Witness: 18 independent points

(41800029847, 4719079210983476)
(45876571447, 5675420192525876)
(30812028847, 3884404658379476)
(-12479014553, 11317972352032676)
(20181196247, 5428367253417076)
(32644665487, -3810579273384364)
(7360873847, -8021938656133324)
(135390399127, -46035550207283404)
(150332668567, 54637052958893876)
(41235033187, -4607646049543564)
(-66023960453, 4537618321160276)
(-19076706553, 12044688521663876)
(56388122263, -8940650106097612)
(30824886247, -3883638204463324)
(-133558553, 9429565645552676)
(-65429995013, 5129270282761136)
(-62531015753, 7229109483649076)
(94911552247, -25046047487098924)

Commentary

Rank-18 naive-height record: h=160.644 (beats prior r18 height record 168.785). Found via a NEW 3-parameter sub-family of Mestre's locus 12*p5=5*p2*p3: the six roots are the union of two depth-2 cubics x^3+px+q, x^3+rx+s with (p-r)(q-s)=0 (here equal product), a slice disjoint from Mestre's (u,v) 2-parameter family. Sextuple {0,54,90,129,585,600}, T=2745/8; certified by injecting the 12 Mestre base points + Neron-Tate height-pairing independence (18 indep).

last edited by RoyManami at 2026-06-25 23:05:55 · history