curve #5692

y² = x³ − 512934393747598419154725812412x + 141257684826737600644594304891041968424197684

a-invariants: [0, 0, 0, -512934393747598419154725812412, 141257684826737600644594304891041968424197684]
c-invariants: [24620850899884724119426838995776, -122046639690301286956929479425860260718506798976]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 57578227304047373592559760860388964753492018986548896974399690484840
discriminant (Δ): 17037168046036058290571260473842969340119655341326726959051214853649141809644348884000000
Faltings height: 15.8432
naive height: 216.8434
primes of bad reduction: 2, 3, 5, 7, 11, 13, 43, 311, 24967, 13694806471, 34943838246788200124719748695571928665263229
regulator: 23860263390399418025.518780012079495589055668548158510369220190716906290639610450
submitted by: Nikita-Shulga
submitted at: 2026-10-24 12:57:42 UTC
last updated: 2026-10-24 12:57:42 UTC

Witness: 21 independent points log in to add more points →

(380713120902273, 1076259745465970390325)
(424644396055038, 126737028326211333030)
(-206967412800990, 15445155698942381670258)
(143246177143228, 8409585545079276547990)
(376368741383508, 1232532194175112017330)
(492149897848698, -2832216645573889254450)
(431180096786598, 503965382843494722150)
(-609991379432562, 15072221675342915110170)
(1228394029902198, -36942653318585589625050)
(424841653995918, 147111588054437716710)
(537107815633548, -4550171076431690960250)
(402841817880804, 6916907771349159870)
(1397993799383817/4, 17304526412121356533125/8)
(7403090815772193/16, 110520016812998567549775/64)
(37505111529548092/9, 7162448496414542364343310/27)
(773296639504548, 14388506352691860138750)
(776514775855978, -14531885599515829534990)
(13752990895803318, 1610711960090258151216090)
(426129516518865, 245986732425134623923)
(1652542869407358, -61697143531183767390630)
(79361805721190964/25, -21831380773801389463735062/125)

Commentary

Specialization T = -106/7 of the elliptic-surface construction associated with curve#3792. Found and verified in an AI-assisted computational search. These 21 rational points certify a rank lower bound of 21, not an exact-rank claim. All primes dividing the minimal discriminant are supplied for conductor computation.

last edited by Nikita-Shulga at 2026-10-24 12:57:43 UTC · history