curve #2733

y² + xy = x³ − 64914943238701568084259068655718987113244635x + 201309851116426413192838237491040059551086270277995189126901980497

a-invariants: [1, 0, 0, -64914943238701568084259068655718987113244635, 201309851116426413192838237491040059551086270277995189126901980497]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 499856691314132909535833670
discriminant (Δ): 8744701223422483493460730533795002513402641431660464141383383853214823438704897030863551986553181615549286556738281250000000000
Faltings height: 23.7246
naive height: 314.2586
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 37, 43, 59, 61, 7589, 44351, 347609
regulator: 10099352.06625868481970059514491910909255060679
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:19:51 UTC
last updated: 2026-09-26 20:19:51 UTC

Witness: 4 independent points log in to add more points →

(4648622968788630263334, 286071443448070484600366165433)
(30853075354468353282483305777490600923154/5593021890913929481, 1392292028331306050080987557715740314945978707215580475861607/13227256602058425487519156379)
(532633553247532550198370542118960808806/7518415477187209, 12216318600514937453986285746226163634686542419749575465509/651912761373083836177573)
(46307090881491983622039471116441582820390282366590063364108321210644313089999312790773286/7175486031580206460405079239751942164909411909778681391199923561529, 4347323589777295927861186486861450646613038830590930371936623757908181387604901438197687845711624313827516136725262606458200403374539/19221044536872879435037625932731924630264359662112071756033859576897923856065573773332973490079352083)

Commentary

2-isogenous image of D-z10-014, rational kernel x=-2324900972751723202914. Source: https://dujella.github.io/tors/z10.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:19:51 UTC · history