curve #91

y² + xy = x³ + x² − 6540850922519953718x − 1851537704759959591812063912

a-invariants: [1, 1, 0, -6540850922519953718, -1851537704759959591812063912]
c-invariants: [313960844280957778489, 1599728574557898755218439881363]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 609139444635528560504687036989987088238968485253777790
discriminant (Δ): 16428490821820205276811409387619951769804980047294386996300
Faltings height: 9.8620
naive height: 141.5874
primes of bad reduction: 2, 3, 5, 29, 31, 3169408342209139, 44658483011160071, 159570870149555903
regulator: 876682647260930.4046834775127923693175750806292138104214702456082460330
submitted by: David Renshaw
submitted at: 2026-06-11 18:54:07 UTC
last updated: 2026-07-01 22:36:42 UTC

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

(-1579605826, 67373394280418)
(-2401956187, -1210893568960)
(-3224222629/4, -430589824752611/8)
(6123597938, -433270141862710)
(-2385452686, 13313205098948)
(-9579433429/4, -70431852761111/8)
(-2399183526, 5592975096618)
(-297102431, 8095815100788)
(-2308393444, 30768491385776)
(-2312819314, -30076910607472)
(3089188419, 86154878976843)
(3789601958, -166687314398248)
(-1454765926, 67713978334268)
(-2193105226, -44101458225592)
(-453816781, 31990018866818)
(-2028432901, 55409236846283)

Commentary

New curve of rank >= 16 and naive height 141.587, found as the specialization T = 482 (t = 241) of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique definie sur Q de rang >= 22' (Acta Arith. 82 (1997), sextuple {0,55,314,378,1007,1036}), located by an exhaustive minimal-height scan crossed with a staged Nagao-sum sieve; 16 independent points certified by positive-definite Neron-Tate height pairing.

last edited by David Renshaw at 2026-06-11 18:54:07 UTC · history