curve #6492

y² + xy = x³ − x² − 237905368786990370844114x + 63251713740961730404610619988195520

a-invariants: [1, -1, 0, -237905368786990370844114, 63251713740961730404610619988195520]
c-invariants: [11419457701775537800517481, -54649480672139547509925585749698600629]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 26334328961446354492261518081380917681712112753779724544284319830
discriminant (Δ): -866564012387594102510980579365440822463839210304064060785355899405937500
Faltings height: 12.5063
naive height: 173.7880
primes of bad reduction: 2, 3, 5, 13, 313, 5580100628223259, 12886946520483517602520702479333533488544497
regulator: 9956358436237454257.54895726260377887093717407568272270911471074905871220310542
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:53:18 UTC
last updated: 2026-11-04 04:53:18 UTC

Witness: 20 independent points log in to add more points →

(27277259374/49, -86173776996088395734/343)
(129397655757086, -1471927952350318832708)
(109766581793344/1369, -10708278553427777609864/50653)
(10277752054264264/6889, -995138151079342935436156/571787)
(627655569844291/441, -14954948984103013116043/9261)
(1467619707276346/2401, -44986651224423911229032/117649)
(3574650810363496/19321, 429438418929977252073628/2685619)
(388279859453299/1369, 6906559061067702727621/50653)
(8060861704918624/24649, 553063665933446434170016/3869893)
(-8683288988459/16, -11541576049989069707/64)
(-662163627094766/5041, -108698595972137713585246/357911)
(-503458984024, 235404335689709812)
(-445892920354, -284040938896999428)
(263556983494989629/3157729, 1176719563136239432300438221/5611284433)
(-1468569612281/4, -2543802760747785649/8)
(-1096876515952848974/5067001, -3688999423907803208594319638/11405819251)
(-257456095898681/3844, -66938026526589268162639/238328)
(30456837860889469/214369, -17843128408934795320181261/99252847)
(-16165447381049/121, -405138071564040335773/1331)
(1719231313751083594/13965169, -9878449682385014825412626914/52187836553)

Commentary

Mestre ten-root construction, roots 0,14,37,83,84,98,139,148,157,180: 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 >= 20: the 20 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 04:53:18 UTC · history