curve #522

y² + xy + y = x³ + x² − 95601193732825621933655670685001977657677704472772663x + 11205917328296652514020167327343969484205598072196150707448747360751344747408781

a-invariants: [1, 1, 1, -95601193732825621933655670685001977657677704472772663, 11205917328296652514020167327343969484205598072196150707448747360751344747408781]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 132427244033169534332852238392107081376038339664432536271108120645163730957223993769897688950
discriminant (Δ): 1672999032487882401600002297691034484154256063982955395004797489537210681366640681503241840336803431618425255372403228999940038083464389977656159178326016000000
Faltings height: 29.3491
naive height: 377.5897
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 31, 61, 233, 521, 601, 757, 797, 977, 2029, 2441, 3001, 5791, 9013, 16981, 32413, 35759, 346361, 393103, 604697, 5987573, 15191747
regulator: 129742692587123759061163116.247289465484688040297560525589595356949177241
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:42:59 UTC
last updated: 2026-09-02 13:42:59 UTC

Witness: 17 independent points log in to add more points →

(301475013734144929167734270442031555/339628041, 156571848725180775412346645722659547050132062742574258/6259005167589)
(6862446495868157160971225548355/22201, 11065018236127042770562741021700489481839504178/3307949)
(-7480147558863083418513373127100251209165/47296278718441, 1538407733793334176676552166391273665199701809205250987566898/325267339594545279989)
(49818206914222594725463042440176008209379/239126790875625, 2128288292844129711985800153111018251614279609495561491385042/3697790934233174203125)
(23582801092373036263526218813158667466891/8298138661201, 3600929996109820568719408022025445583240402157291836385350250/23904024836249999449)
(16293445876265277865265130202472785357924644605/15933405766205695681, 1993841008108400901236071964671081548978392759720267650402152129243312/63600850647842627076323749921)
(-24279496607596246930300339566379067814958957230786565/68868132131682223518516529, 596952991795818124967849745901379252668439776210586706625076536823148964951386/571514766646174131946012924032009185833)
(3453865162765215437356618723109259926978033571195/26823623085900176260729, 4460601153853292390066058860792056162926577181023718822460606437916250126/4393151262598269112158729188549683)
(107620539733184441266757076261705690171352736455/1170828903405166401796, 71610731799300798843025303380306704106596680265023480467762660911708793/40062694240868547846692819076344)
(-133059344911890628303907053175610107505372685/826984604823789721, 3561139508482556481434250931273333527710693528922740704900385055198/752048998380830768759854019)
(7621158304514618590206491077052398197859/4016549081689, 656981366147489724908272337425764548158777672928423937267330/8049698560732620637)
(4131119674699869880279350340467812483303306115/51806408151464521, 265520712280085752117529130046217053801281780354848255274759658805902/11791667490506959583823931)
(10895841283945353202318821877239270157851559305241835/405228154309545511201121089, 23998181861667877805864792611106084378168261246730565495409480202333914041752054/8157356023236244167036051795827454694687)
(12178100143627147345907656830050848835693755/115984002247906081, 60236348308121718220062684979170837060669344873869662838626462362/39500003693700428315049521)
(17014001794362800144846696773899720564746795/41023021549539649, 54419274069746417928867197799242117112583644999167473686787133046/8308860504035080710746143)
(59241093984320983764046395387906939372865/4157590326289, 14415614974353020206567342241429351555748280955599987973998516/8477397354338817913)
(-9043959272602262167324615965877355285/55722907249, 62320114739186389786449298222353778136191280642112823226/13153782316477193)

Commentary

Exact rank 17, unconditional: rational 2-torsion point, 17 independent points (Neron-Tate height matrix of rank 17; the archived certificate additionally saturates the lattice at all primes up to 97 — the points listed here are the ellrank output, not the saturated basis) with matching 2-descent upper bound 17 (PARI/GP 2.15.4 ellrank, effort 2). Fiber t=106/7989 on branch u=58/37 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found by a Mestre-Nagao residue-table sieve followed by an exact 2-descent-type upper bound and ellrank (Matthias Breddin, 2026-09-02; method: doi:10.5281/zenodo.22242107).

last edited by Matthias Breddin at 2026-09-02 13:42:59 UTC · history