Growth rates of hyperbolic Coxeter triangle groups

diagram

$\tau$

[7, 3]:

1.176280818259917506544070338474035050693415806564695259830106347029688376548549962096830115581815395

comment: $\Delta(2,3,7)$. The minimal polynomial of $\tau$ is $x^{10} + x^{9} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} + x + 1$. This is Lehmer's number.

[8, 3]:

1.230391434407224702790177938975279017566574489661756241401914236172813447853545416735984651662408529

comment: $\Delta(2,3,8)$. The minimal polynomial of $\tau$ is $x^{10} - x^{7} - x^{5} - x^{3} + 1$.

[9, 3]:

1.261230961137138851946671503074483335906561512823932899174657356830904239469104488591116935467487797

comment: $\Delta(2,3,9)$. The minimal polynomial of $\tau$ is $x^{10} - x^{8} - x^{5} - x^{2} + 1$.

[10, 3]:

1.280638156267757596701902532710676301595313317819353584076277261716335555633386491878399947424007022

comment: $\Delta(2,3,10)$. The minimal polynomial of $\tau$ is $x^{8} - x^{5} - x^{4} - x^{3} + 1$.

[11, 3]:

1.293485953125454106519909883794095300396496621900788800582481240648323409236024590193726706779689885

comment: $\Delta(2,3,11)$. The minimal polynomial of $\tau$ is $x^{10} - x^{8} - x^{7} + x^{5} - x^{3} - x^{2} + 1$.

[12, 3]:

1.302268805094334462964240864634518783740979954013886025850146135774028019627834630123986977095717814

comment: $\Delta(2,3,12)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{7} + x^{6} - x^{5} - x + 1$.

[5, 4]:

1.280638156267757596701902532710676301595313317819353584076277261716335555633386491878399947424007022

comment: $\Delta(2,4,5)$. The minimal polynomial of $\tau$ is $x^{8} - x^{5} - x^{4} - x^{3} + 1$.

[6, 4]:

1.359999711711500865451030495300356648713475167436404678928709599000918565827376447822831143284463018

comment: $\Delta(2,4,6)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} + x^{6} - 2 x^{5} + x^{4} - 2 x^{3} + x^{2} - x + 1$.

[7, 4]:

1.401268367939854915101764095621406049201490677870198100145488730144935422455428299191458403803013649

comment: $\Delta(2,4,7)$. The minimal polynomial of $\tau$ is $x^{6} - x^{4} - x^{3} - x^{2} + 1$.

[8, 4]:

1.425005267838192556664667892955431790335881960629575792302227494319146499713934419442899032862339456

comment: $\Delta(2,4,8)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} - x^{5} + x^{4} - x^{3} - x + 1$.

[9, 4]:

1.439398612809279305159471019619134686011106438895082030006345601361074905569152485039090235144942600

comment: $\Delta(2,4,9)$. The minimal polynomial of $\tau$ is $x^{12} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} + 1$.

[10, 4]:

1.448423040244205801526893998204145083563506879913308068612936614937762577778546801592194681320894080

comment: $\Delta(2,4,10)$. The minimal polynomial of $\tau$ is $x^{10} - 2 x^{9} + 2 x^{8} - 2 x^{7} + x^{6} - x^{5} + x^{4} - 2 x^{3} + 2 x^{2} - 2 x + 1$.

[11, 4]:

1.454212370577025938845183383790126631147859638547266641684649875894264642217768737043738253729673538

comment: $\Delta(2,4,11)$. The minimal polynomial of $\tau$ is $x^{14} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} + 1$.

[12, 4]:

1.457987479648116075716627246227500805053291692966023399159586536681253211620807327451670034446778139

comment: $\Delta(2,4,12)$. The minimal polynomial of $\tau$ is $x^{8} - x^{6} - x^{5} - x^{3} - x^{2} + 1$.

[5, 5]:

1.401268367939854915101764095621406049201490677870198100145488730144935422455428299191458403803013649

comment: $\Delta(2,5,5)$. The minimal polynomial of $\tau$ is $x^{6} - x^{4} - x^{3} - x^{2} + 1$.

[6, 5]:

1.457987479648116075716627246227500805053291692966023399159586536681253211620807327451670034446778139

comment: $\Delta(2,5,6)$. The minimal polynomial of $\tau$ is $x^{8} - x^{6} - x^{5} - x^{3} - x^{2} + 1$.

[7, 5]:

1.488565327751942904466697019643593494101637809802461928291759314368766127207409172721877431729369573

comment: $\Delta(2,5,7)$. The minimal polynomial of $\tau$ is $x^{12} + x^{11} - x^{9} - 2 x^{8} - 3 x^{7} - 3 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} + x + 1$.

[8, 5]:

1.506135679553838823954261403643581593816215611682478824215182992478173582486762058875853445084229649

comment: $\Delta(2,5,8)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{3} - x + 1$.

[9, 5]:

1.516629599519131098775517547685850545599288741924357749430398187822407130434195037753625225303367067

comment: $\Delta(2,5,9)$. The minimal polynomial of $\tau$ is $x^{14} + x^{13} - x^{11} - 2 x^{10} - 3 x^{9} - 3 x^{8} - 3 x^{7} - 3 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} + x + 1$.

[10, 5]:

1.523060248947063274836994929362511652733272706765376559800731245888613837170145454437921548126141118

comment: $\Delta(2,5,10)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} - x^{6} + x^{4} - x^{2} - x + 1$.

[11, 5]:

1.527071755783427413658996991475831979164091733954163194522322310186512162634206616585411740273153009

comment: $\Delta(2,5,11)$. The minimal polynomial of $\tau$ is $x^{16} + x^{15} - x^{13} - 2 x^{12} - 3 x^{11} - 3 x^{10} - 3 x^{9} - 3 x^{8} - 3 x^{7} - 3 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} + x + 1$.

[12, 5]:

1.529605902347355366405913969464521879526582482335190118628327237497644649407276887270206672272446593

comment: $\Delta(2,5,12)$. The minimal polynomial of $\tau$ is $x^{16} - x^{13} - x^{12} - 2 x^{11} - x^{10} - 2 x^{9} - x^{8} - 2 x^{7} - x^{6} - 2 x^{5} - x^{4} - x^{3} + 1$.

[6, 6]:

1.506135679553838823954261403643581593816215611682478824215182992478173582486762058875853445084229649

comment: $\Delta(2,6,6)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{3} - x + 1$.

[7, 6]:

1.532257559331558681570239081113132998342063455128465272644490341352345292361577638793108054268951196

comment: $\Delta(2,6,7)$. The minimal polynomial of $\tau$ is $x^{12} - x^{9} - x^{8} - 2 x^{7} - 2 x^{6} - 2 x^{5} - x^{4} - x^{3} + 1$.

[8, 6]:

1.547196965626387186895002889439719134824647895007838988713924997424884815656829457058653746949595704

comment: $\Delta(2,6,8)$. The minimal polynomial of $\tau$ is $x^{8} - 2 x^{7} + 2 x^{6} - 3 x^{5} + 3 x^{4} - 3 x^{3} + 2 x^{2} - 2 x + 1$.

[9, 6]:

1.556030191322682260530428926604570466321314155677330741391368526702608303524291254496090809974014656

comment: $\Delta(2,6,9)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{4} + x^{3} - x^{2} - x + 1$.

[10, 6]:

1.561371339437330527455747152663065652473477936488937868392344623642407679409851848740766966140539883

comment: $\Delta(2,6,10)$. The minimal polynomial of $\tau$ is $x^{14} - x^{13} + x^{12} - 2 x^{11} + x^{10} - 3 x^{9} + x^{8} - 3 x^{7} + x^{6} - 3 x^{5} + x^{4} - 2 x^{3} + x^{2} - x + 1$.

[11, 6]:

1.564651437616732085454745564587759098946411028375401049213862741761309358706147021818172165002582328

comment: $\Delta(2,6,11)$. The minimal polynomial of $\tau$ is $x^{16} - x^{13} - x^{12} - 2 x^{11} - 2 x^{10} - 2 x^{9} - 2 x^{8} - 2 x^{7} - 2 x^{6} - 2 x^{5} - x^{4} - x^{3} + 1$.

[12, 6]:

1.566687843732119566585865138776246563196284770332494694131629807932762379413391528068552477392901789

comment: $\Delta(2,6,12)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{9} - x^{7} + x^{6} - x^{5} - x^{3} - x + 1$.

[7, 7]:

1.556030191322682260530428926604570466321314155677330741391368526702608303524291254496090809974014656

comment: $\Delta(2,7,7)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{4} + x^{3} - x^{2} - x + 1$.

[8, 7]:

1.569589535301095622500619421678559442097016570801706727438181789566266507232095730447984876835103190

comment: $\Delta(2,7,8)$. The minimal polynomial of $\tau$ is $x^{14} - x^{11} - x^{10} - 2 x^{9} - 2 x^{8} - 3 x^{7} - 2 x^{6} - 2 x^{5} - x^{4} - x^{3} + 1$.

[9, 7]:

1.577562082409277645454634973649404976546931276958939592560447037380023403956573035963012580967249743

comment: $\Delta(2,7,9)$. The minimal polynomial of $\tau$ is $x^{16} + x^{15} - x^{13} - 2 x^{12} - 3 x^{11} - 4 x^{10} - 5 x^{9} - 5 x^{8} - 5 x^{7} - 4 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} + x + 1$.

[10, 7]:

1.582347183680458215564288456693430845102163306325701082519164370447299929962685488727640295954001468

comment: $\Delta(2,7,10)$. The minimal polynomial of $\tau$ is $x^{6} - x^{4} - 2 x^{3} - x^{2} + 1$.

[11, 7]:

1.585260364361682930867475471345107339219847862656400302755617893199597776396960402322569197437320541

comment: $\Delta(2,7,11)$. The minimal polynomial of $\tau$ is $x^{18} + x^{17} - x^{15} - 2 x^{14} - 3 x^{13} - 4 x^{12} - 5 x^{11} - 5 x^{10} - 5 x^{9} - 5 x^{8} - 5 x^{7} - 4 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} + x + 1$.

[12, 7]:

1.587051626553847407642002775516578718088356879884748045908686186105024470541927734151296378176713520

comment: $\Delta(2,7,12)$. The minimal polynomial of $\tau$ is $x^{18} - x^{15} - x^{14} - 2 x^{13} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} - 2 x^{8} - 3 x^{7} - 2 x^{6} - 2 x^{5} - x^{4} - x^{3} + 1$.

[8, 8]:

1.582347183680458215564288456693430845102163306325701082519164370447299929962685488727640295954001468

comment: $\Delta(2,8,8)$. The minimal polynomial of $\tau$ is $x^{6} - x^{4} - 2 x^{3} - x^{2} + 1$.

[9, 8]:

1.589826164371655892980081517890299973147767553902382894228811868736835449755316948130324178990749970

comment: $\Delta(2,8,9)$. The minimal polynomial of $\tau$ is $x^{16} - x^{13} - x^{12} - 2 x^{11} - 2 x^{10} - 3 x^{9} - 3 x^{8} - 3 x^{7} - 2 x^{6} - 2 x^{5} - x^{4} - x^{3} + 1$.

[10, 8]:

1.594296644465077363260638507918967086881717971601264811336503154283569177054233114701417353999095736

comment: $\Delta(2,8,10)$. The minimal polynomial of $\tau$ is $x^{16} - x^{15} + x^{14} - 2 x^{13} + x^{12} - 3 x^{11} + x^{10} - 4 x^{9} + x^{8} - 4 x^{7} + x^{6} - 3 x^{5} + x^{4} - 2 x^{3} + x^{2} - x + 1$.

[11, 8]:

1.597005009172732477134843928895980336464006312033487359865843548509248356063757479301902268580260675

comment: $\Delta(2,8,11)$. The minimal polynomial of $\tau$ is $x^{10} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} + 1$.

[12, 8]:

1.598661268392370352387858142892685568892388823457201328385952749863444311025646221545859684251676062

comment: $\Delta(2,8,12)$. The minimal polynomial of $\tau$ is $x^{10} - 2 x^{9} + x^{8} - x^{7} + 2 x^{6} - 3 x^{5} + 2 x^{4} - x^{3} + x^{2} - 2 x + 1$.

[9, 9]:

1.597005009172732477134843928895980336464006312033487359865843548509248356063757479301902268580260675

comment: $\Delta(2,9,9)$. The minimal polynomial of $\tau$ is $x^{10} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} + 1$.

[10, 9]:

1.601286383464232555680387515149525599657382537277038264213334252717112525807684761333030710922260355

comment: $\Delta(2,9,10)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - x^{13} - x^{11} - 2 x^{10} - 2 x^{9} - x^{8} - 2 x^{7} - 2 x^{6} - x^{5} - x^{3} - x^{2} + 1$.

[11, 9]:

1.603872989503687182789025224772083016539647290371039786877268816773404652170859190883620311563013026

comment: $\Delta(2,9,11)$. The minimal polynomial of $\tau$ is $x^{20} + x^{19} - x^{17} - 2 x^{16} - 3 x^{15} - 4 x^{14} - 5 x^{13} - 6 x^{12} - 7 x^{11} - 7 x^{10} - 7 x^{9} - 6 x^{8} - 5 x^{7} - 4 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} + x + 1$.

[12, 9]:

1.605449828807924074566668128966985826547173713641293500068747478798132129837717517988576647193612139

comment: $\Delta(2,9,12)$. The minimal polynomial of $\tau$ is $x^{8} - 2 x^{7} + x^{6} - x^{4} + x^{2} - 2 x + 1$.

[10, 10]:

1.605449828807924074566668128966985826547173713641293500068747478798132129837717517988576647193612139

comment: $\Delta(2,10,10)$. The minimal polynomial of $\tau$ is $x^{8} - 2 x^{7} + x^{6} - x^{4} + x^{2} - 2 x + 1$.

[11, 10]:

1.607961206702966786199098501246180781514441404265107877636237590674983304397149891934696945135197246

comment: $\Delta(2,10,11)$. The minimal polynomial of $\tau$ is $x^{16} - x^{13} - 2 x^{12} - 2 x^{11} - 2 x^{10} - 2 x^{9} - x^{8} - 2 x^{7} - 2 x^{6} - 2 x^{5} - 2 x^{4} - x^{3} + 1$.

[12, 10]:

1.609489393085513742703021015312050161414489094937984593831703190731527164517161857273974971131436497

comment: $\Delta(2,10,12)$. The minimal polynomial of $\tau$ is $x^{20} - x^{19} + x^{18} - 2 x^{17} + x^{16} - 3 x^{15} + x^{14} - 4 x^{13} + x^{12} - 5 x^{11} + x^{10} - 5 x^{9} + x^{8} - 4 x^{7} + x^{6} - 3 x^{5} + x^{4} - 2 x^{3} + x^{2} - x + 1$.

[11, 11]:

1.610424982281390207266099303335445990244075049184095514567471479637677922024302567977790732236781987

comment: $\Delta(2,11,11)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} + 1$.

[12, 11]:

1.611922603419819362624569737497505286973798729662678237494026789481208579146417535441937900713759122

comment: $\Delta(2,11,12)$. The minimal polynomial of $\tau$ is $x^{22} - x^{19} - x^{18} - 2 x^{17} - 2 x^{16} - 3 x^{15} - 3 x^{14} - 4 x^{13} - 4 x^{12} - 5 x^{11} - 4 x^{10} - 4 x^{9} - 3 x^{8} - 3 x^{7} - 2 x^{6} - 2 x^{5} - x^{4} - x^{3} + 1$.

[12, 12]:

1.613400721392476208009183359292431274268671286375351140260565585503304105304832988630079986522083320

comment: $\Delta(2,12,12)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{9} - x^{7} - x^{5} - x^{3} - x + 1$.

[(3, 3, 4)]:

1.401268367939854915101764095621406049201490677870198100145488730144935422455428299191458403803013649

comment: $\Delta(3,3,4)$. The minimal polynomial of $\tau$ is $x^{6} - x^{4} - x^{3} - x^{2} + 1$.

[(3, 3, 5)]:

1.506135679553838823954261403643581593816215611682478824215182992478173582486762058875853445084229649

comment: $\Delta(3,3,5)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{3} - x + 1$.

[(3, 3, 6)]:

1.556030191322682260530428926604570466321314155677330741391368526702608303524291254496090809974014656

comment: $\Delta(3,3,6)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{4} + x^{3} - x^{2} - x + 1$.

[(3, 3, 7)]:

1.582347183680458215564288456693430845102163306325701082519164370447299929962685488727640295954001468

comment: $\Delta(3,3,7)$. The minimal polynomial of $\tau$ is $x^{6} - x^{4} - 2 x^{3} - x^{2} + 1$.

[(3, 3, 8)]:

1.597005009172732477134843928895980336464006312033487359865843548509248356063757479301902268580260675

comment: $\Delta(3,3,8)$. The minimal polynomial of $\tau$ is $x^{10} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} + 1$.

[(3, 3, 9)]:

1.605449828807924074566668128966985826547173713641293500068747478798132129837717517988576647193612139

comment: $\Delta(3,3,9)$. The minimal polynomial of $\tau$ is $x^{8} - 2 x^{7} + x^{6} - x^{4} + x^{2} - 2 x + 1$.

[(3, 3, 10)]:

1.610424982281390207266099303335445990244075049184095514567471479637677922024302567977790732236781987

comment: $\Delta(3,3,10)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} + 1$.

[(3, 3, 11)]:

1.613400721392476208009183359292431274268671286375351140260565585503304105304832988630079986522083320

comment: $\Delta(3,3,11)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{9} - x^{7} - x^{5} - x^{3} - x + 1$.

[(3, 3, 12)]:

1.615199132089904938531648866253306106914859540142844055080065042595008740735102024754474130884835366

comment: $\Delta(3,3,12)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} + x^{9} - x^{8} - x^{7} + x^{6} - x^{5} - x^{4} + x^{3} - x^{2} - x + 1$.

[(3, 4, 4)]:

1.582347183680458215564288456693430845102163306325701082519164370447299929962685488727640295954001468

comment: $\Delta(3,4,4)$. The minimal polynomial of $\tau$ is $x^{6} - x^{4} - 2 x^{3} - x^{2} + 1$.

[(3, 4, 5)]:

1.657403502227057448215922702877809947391064966836956284148611218263876463661033238698691382573205955

comment: $\Delta(3,4,5)$. The minimal polynomial of $\tau$ is $x^{10} + x^{9} - 2 x^{7} - 4 x^{6} - 5 x^{5} - 4 x^{4} - 2 x^{3} + x + 1$.

[(3, 4, 6)]:

1.693507382606863895990666092145490252460203155836634649373199800822992479019388273959736990537359533

comment: $\Delta(3,4,6)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} - x^{5} - x^{4} - x^{3} - x + 1$.

[(3, 4, 7)]:

1.712107608534026699629771845067319637729436592554209931494715954739179807031522479309503428535473833

comment: $\Delta(3,4,7)$. The minimal polynomial of $\tau$ is $x^{12} + x^{11} - 2 x^{9} - 4 x^{8} - 5 x^{7} - 5 x^{6} - 5 x^{5} - 4 x^{4} - 2 x^{3} + x + 1$.

[(3, 4, 8)]:

1.722083805739042245027069212153831462070116555751550307048783133542303795066098290709443333736129248

comment: $\Delta(3,4,8)$. The minimal polynomial of $\tau$ is $x^{4} - x^{3} - x^{2} - x + 1$.

[(3, 4, 9)]:

1.727572048697291555072037230667696170222406262238714600072486932498070490757362384668394570598825521

comment: $\Delta(3,4,9)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - x^{9} - 2 x^{8} - 2 x^{7} - x^{6} - 2 x^{5} - 2 x^{4} - x^{3} - x^{2} + 1$.

[(3, 4, 10)]:

1.730641327568503042476643295248103052824147938615575053905177984330670908544094076016262386986336775

comment: $\Delta(3,4,10)$. The minimal polynomial of $\tau$ is $x^{14} - 2 x^{11} - 2 x^{10} - 3 x^{9} - 2 x^{8} - 3 x^{7} - 2 x^{6} - 3 x^{5} - 2 x^{4} - 2 x^{3} + 1$.

[(3, 4, 11)]:

1.732376319923863712677336110829659088168372220910774083531167246929717509037224597549587603660414697

comment: $\Delta(3,4,11)$. The minimal polynomial of $\tau$ is $x^{16} + x^{15} - 2 x^{13} - 4 x^{12} - 5 x^{11} - 5 x^{10} - 5 x^{9} - 5 x^{8} - 5 x^{7} - 5 x^{6} - 5 x^{5} - 4 x^{4} - 2 x^{3} + x + 1$.

[(3, 4, 12)]:

1.733363963620486780060514473137713377222897366413169626676313550817151026293544018959111646020768304

comment: $\Delta(3,4,12)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} - x^{7} - x^{5} - x^{2} - x + 1$.

[(3, 5, 5)]:

1.722083805739042245027069212153831462070116555751550307048783133542303795066098290709443333736129248

comment: $\Delta(3,5,5)$. The minimal polynomial of $\tau$ is $x^{4} - x^{3} - x^{2} - x + 1$.

[(3, 5, 6)]:

1.753097478214463946142800254714498521916830531447376275126367461837846776964046144160281695000101052

comment: $\Delta(3,5,6)$. The minimal polynomial of $\tau$ is $x^{10} - x^{8} - x^{7} - 2 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} - x^{2} + 1$.

[(3, 5, 7)]:

1.768866857037396650847876949061289651787292618880690279179247406315857654779606954498655378542399075

comment: $\Delta(3,5,7)$. The minimal polynomial of $\tau$ is $x^{12} - 2 x^{9} - 2 x^{8} - 4 x^{7} - 3 x^{6} - 4 x^{5} - 2 x^{4} - 2 x^{3} + 1$.

[(3, 5, 8)]:

1.777171932091800169133125651027771569483247875177801146966221146215787512932182968077768951183086051

comment: $\Delta(3,5,8)$. The minimal polynomial of $\tau$ is $x^{14} + x^{13} - 2 x^{11} - 4 x^{10} - 6 x^{9} - 7 x^{8} - 7 x^{7} - 7 x^{6} - 6 x^{5} - 4 x^{4} - 2 x^{3} + x + 1$.

[(3, 5, 9)]:

1.781643598608001947392663351969728057845449665336420046019487278499963638093412619247397379694995723

comment: $\Delta(3,5,9)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{4} - x^{2} - x + 1$.

[(3, 5, 10)]:

1.784085467642839633384150352349398546056563821173208795418225611010451140402784359822906021792023874

comment: $\Delta(3,5,10)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - 2 x^{9} - 2 x^{8} - x^{7} - x^{6} - x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(3, 5, 11)]:

1.785431006965268249431687740933819987527259960175728404736433434600719488630299630833801047141769898

comment: $\Delta(3,5,11)$. The minimal polynomial of $\tau$ is $x^{16} - 2 x^{13} - 2 x^{12} - 4 x^{11} - 3 x^{10} - 4 x^{9} - 3 x^{8} - 4 x^{7} - 3 x^{6} - 4 x^{5} - 2 x^{4} - 2 x^{3} + 1$.

[(3, 5, 12)]:

1.786176717177689247844351456550960387495053804529216223875965962179069759412486191083548531896942484

comment: $\Delta(3,5,12)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - x^{13} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 2 x^{9} - 3 x^{8} - 2 x^{7} - 2 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} - x^{2} + 1$.

[(3, 6, 6)]:

1.781643598608001947392663351969728057845449665336420046019487278499963638093412619247397379694995723

comment: $\Delta(3,6,6)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{4} - x^{2} - x + 1$.

[(3, 6, 7)]:

1.796072323163702668114467413935195800173287708068259341418284202581710051871987650612114013382238906

comment: $\Delta(3,6,7)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} - x^{6} - x^{4} - x^{2} - x + 1$.

[(3, 6, 8)]:

1.803606417529837285932282990038703108542252955100567473170568570909194698680254087109533071730796883

comment: $\Delta(3,6,8)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{9} - x^{8} - 2 x^{7} - x^{6} - 2 x^{5} - x^{4} - x^{3} - x + 1$.

[(3, 6, 9)]:

1.807621596209608786244488407237222356027382481561657947873209950918659741713266178835655248278904137

comment: $\Delta(3,6,9)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} + x^{9} - 2 x^{8} - 2 x^{7} + x^{6} - 2 x^{5} - 2 x^{4} + x^{3} - x^{2} - x + 1$.

[(3, 6, 10)]:

1.809789333465951272221051924878982619820568610713602743429233481875512248033360339745399933385189736

comment: $\Delta(3,6,10)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} - 2 x^{5} - 2 x^{3} - x + 1$.

[(3, 6, 11)]:

1.810969317457893000398089526804043781008789396904930030342459304877879690774205492608178101712537191

comment: $\Delta(3,6,11)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - x^{13} - 2 x^{12} - 3 x^{11} - 3 x^{10} - 3 x^{9} - 3 x^{8} - 3 x^{7} - 3 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} - x^{2} + 1$.

[(3, 6, 12)]:

1.811614963022274584850478493316246324186359537073618196546719533282930534731827738794221530604781780

comment: $\Delta(3,6,12)$. The minimal polynomial of $\tau$ is $x^{8} - 2 x^{7} + x^{5} - x^{4} + x^{3} - 2 x + 1$.

[(3, 7, 7)]:

1.809789333465951272221051924878982619820568610713602743429233481875512248033360339745399933385189736

comment: $\Delta(3,7,7)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} - 2 x^{5} - 2 x^{3} - x + 1$.

[(3, 7, 8)]:

1.816922832018760342925771306925039936353156278417472833832546179670023799475876440377413371025090096

comment: $\Delta(3,7,8)$. The minimal polynomial of $\tau$ is $x^{16} + x^{15} - 2 x^{13} - 4 x^{12} - 6 x^{11} - 8 x^{10} - 10 x^{9} - 11 x^{8} - 10 x^{7} - 8 x^{6} - 6 x^{5} - 4 x^{4} - 2 x^{3} + x + 1$.

[(3, 7, 9)]:

1.820705505614074607154334713760663282725587692250647661717938297799744716667873575130195624185357928

comment: $\Delta(3,7,9)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - 2 x^{9} - 2 x^{8} - 2 x^{7} - x^{6} - 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(3, 7, 10)]:

1.822736206966662844865497157176384788283139219313287920353365837343243894794425707926530075605977025

comment: $\Delta(3,7,10)$. The minimal polynomial of $\tau$ is $x^{18} + x^{17} - 2 x^{15} - 4 x^{14} - 6 x^{13} - 8 x^{12} - 10 x^{11} - 11 x^{10} - 11 x^{9} - 11 x^{8} - 10 x^{7} - 8 x^{6} - 6 x^{5} - 4 x^{4} - 2 x^{3} + x + 1$.

[(3, 7, 11)]:

1.823834886362848632586234047654783925468235181174710013190922037538678174823961164952983828663573455

comment: $\Delta(3,7,11)$. The minimal polynomial of $\tau$ is $x^{10} - x^{8} - 2 x^{7} - 2 x^{6} - 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(3, 7, 12)]:

1.824432214867466893935561499564098192790219574848914807480443207866078730058019801524189730637202153

comment: $\Delta(3,7,12)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - x^{15} - 2 x^{14} - 3 x^{13} - 3 x^{12} - 4 x^{11} - 4 x^{10} - 3 x^{9} - 4 x^{8} - 4 x^{7} - 3 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} - x^{2} + 1$.

[(3, 8, 8)]:

1.823834886362848632586234047654783925468235181174710013190922037538678174823961164952983828663573455

comment: $\Delta(3,8,8)$. The minimal polynomial of $\tau$ is $x^{10} - x^{8} - 2 x^{7} - 2 x^{6} - 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(3, 8, 9)]:

1.827491034931104200341741409540205069198102372952724792098537664468067084910657148315720340567069312

comment: $\Delta(3,8,9)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - x^{13} - 2 x^{12} - 3 x^{11} - 3 x^{10} - 4 x^{9} - 5 x^{8} - 4 x^{7} - 3 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} - x^{2} + 1$.

[(3, 8, 10)]:

1.829448199563050426193108138839401762972593522226616771987451664086578528470243530021906595154111703

comment: $\Delta(3,8,10)$. The minimal polynomial of $\tau$ is $x^{18} - 2 x^{15} - 2 x^{14} - 4 x^{13} - 4 x^{12} - 6 x^{11} - 6 x^{10} - 7 x^{9} - 6 x^{8} - 6 x^{7} - 4 x^{6} - 4 x^{5} - 2 x^{4} - 2 x^{3} + 1$.

[(3, 8, 11)]:

1.830503799713479414296762378941451534367749318688544511039944527979162385506831036199947691498690469

comment: $\Delta(3,8,11)$. The minimal polynomial of $\tau$ is $x^{14} - x^{12} - 2 x^{11} - 2 x^{10} - 2 x^{9} - 2 x^{8} - x^{7} - 2 x^{6} - 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(3, 8, 12)]:

1.831075825102305589979153880392949852143720246151911007249752104536765546433951371883083737878916508

comment: $\Delta(3,8,12)$. The minimal polynomial of $\tau$ is $x^{6} - 2 x^{5} + x^{3} - 2 x + 1$.

[(3, 9, 9)]:

1.831075825102305589979153880392949852143720246151911007249752104536765546433951371883083737878916508

comment: $\Delta(3,9,9)$. The minimal polynomial of $\tau$ is $x^{6} - 2 x^{5} + x^{3} - 2 x + 1$.

[(3, 9, 10)]:

1.832991976179982579059796490386282234533684699545969763596666059644725015222355701557341107501262273

comment: $\Delta(3,9,10)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - x^{15} - 2 x^{14} - 3 x^{13} - 3 x^{12} - 4 x^{11} - 5 x^{10} - 5 x^{9} - 5 x^{8} - 4 x^{7} - 3 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} - x^{2} + 1$.

[(3, 9, 11)]:

1.834023784630616592212772378590278555433432441328606429141676733568825018621459691845429283618776380

comment: $\Delta(3,9,11)$. The minimal polynomial of $\tau$ is $x^{18} - x^{17} - x^{15} - x^{14} - 2 x^{13} - x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} - 2 x^{8} - 3 x^{7} - x^{6} - 2 x^{5} - x^{4} - x^{3} - x + 1$.

[(3, 9, 12)]:

1.834581958146143559615315163910275044659496517904224092367165299136587918109491196890389518716595692

comment: $\Delta(3,9,12)$. The minimal polynomial of $\tau$ is $x^{18} - x^{17} - x^{16} + x^{15} - 2 x^{14} - 2 x^{13} + x^{12} - 3 x^{11} - 3 x^{10} + x^{9} - 3 x^{8} - 3 x^{7} + x^{6} - 2 x^{5} - 2 x^{4} + x^{3} - x^{2} - x + 1$.

[(3, 10, 10)]:

1.834884773004843595375674284340778292429050701425126987453294763024584204464532397369327489714492059

comment: $\Delta(3,10,10)$. The minimal polynomial of $\tau$ is $x^{8} - x^{6} - 2 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(3, 10, 11)]:

1.835903141927826908674659869881272876420725801865407335685960208247229084801344738372822312513544259

comment: $\Delta(3,10,11)$. The minimal polynomial of $\tau$ is $x^{22} + x^{21} - 2 x^{19} - 4 x^{18} - 6 x^{17} - 8 x^{16} - 10 x^{15} - 12 x^{14} - 14 x^{13} - 16 x^{12} - 17 x^{11} - 16 x^{10} - 14 x^{9} - 12 x^{8} - 10 x^{7} - 8 x^{6} - 6 x^{5} - 4 x^{4} - 2 x^{3} + x + 1$.

[(3, 10, 12)]:

1.836453545686090131005690695312162602729034161484955617155879998835424138202516237982178083536801433

comment: $\Delta(3,10,12)$. The minimal polynomial of $\tau$ is $x^{20} - x^{19} - x^{17} - x^{16} - 2 x^{15} - x^{14} - 3 x^{13} - 2 x^{12} - 3 x^{11} - 3 x^{10} - 3 x^{9} - 2 x^{8} - 3 x^{7} - x^{6} - 2 x^{5} - x^{4} - x^{3} - x + 1$.

[(3, 11, 11)]:

1.836913829152166750480623636182143268557465543464201256623887038628526071822752961267688832855706972

comment: $\Delta(3,11,11)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - 2 x^{9} - 2 x^{7} - 2 x^{5} - 2 x^{3} - x + 1$.

[(3, 11, 12)]:

1.837459817961327519502990896542083914736072039745581970558787721832728858049766670160715917433303085

comment: $\Delta(3,11,12)$. The minimal polynomial of $\tau$ is $x^{18} - x^{17} - x^{16} - x^{14} - x^{12} - 2 x^{11} - x^{10} - x^{9} - x^{8} - 2 x^{7} - x^{6} - x^{4} - x^{2} - x + 1$.

[(3, 12, 12)]:

1.838003282140456612366943118811165813347566623056744798603563278778813499011608291791663392415420349

comment: $\Delta(3,12,12)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} - x^{8} - x^{7} - x^{5} - x^{4} - x^{2} - x + 1$.

[(4, 4, 4)]:

1.722083805739042245027069212153831462070116555751550307048783133542303795066098290709443333736129248

comment: $\Delta(4,4,4)$. The minimal polynomial of $\tau$ is $x^{4} - x^{3} - x^{2} - x + 1$.

[(4, 4, 5)]:

1.781643598608001947392663351969728057845449665336420046019487278499963638093412619247397379694995723

comment: $\Delta(4,4,5)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{4} - x^{2} - x + 1$.

[(4, 4, 6)]:

1.809789333465951272221051924878982619820568610713602743429233481875512248033360339745399933385189736

comment: $\Delta(4,4,6)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} - 2 x^{5} - 2 x^{3} - x + 1$.

[(4, 4, 7)]:

1.823834886362848632586234047654783925468235181174710013190922037538678174823961164952983828663573455

comment: $\Delta(4,4,7)$. The minimal polynomial of $\tau$ is $x^{10} - x^{8} - 2 x^{7} - 2 x^{6} - 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(4, 4, 8)]:

1.831075825102305589979153880392949852143720246151911007249752104536765546433951371883083737878916508

comment: $\Delta(4,4,8)$. The minimal polynomial of $\tau$ is $x^{6} - 2 x^{5} + x^{3} - 2 x + 1$.

[(4, 4, 9)]:

1.834884773004843595375674284340778292429050701425126987453294763024584204464532397369327489714492059

comment: $\Delta(4,4,9)$. The minimal polynomial of $\tau$ is $x^{8} - x^{6} - 2 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(4, 4, 10)]:

1.836913829152166750480623636182143268557465543464201256623887038628526071822752961267688832855706972

comment: $\Delta(4,4,10)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - 2 x^{9} - 2 x^{7} - 2 x^{5} - 2 x^{3} - x + 1$.

[(4, 4, 11)]:

1.838003282140456612366943118811165813347566623056744798603563278778813499011608291791663392415420349

comment: $\Delta(4,4,11)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} - x^{8} - x^{7} - x^{5} - x^{4} - x^{2} - x + 1$.

[(4, 4, 12)]:

1.838591111575771236549450092206018563602108930419563822671624382659166741430796448250416380684298882

comment: $\Delta(4,4,12)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} - x^{9} + x^{8} - x^{7} - x^{6} - x^{5} + x^{4} - x^{3} - x^{2} - x + 1$.

[(4, 5, 5)]:

1.834884773004843595375674284340778292429050701425126987453294763024584204464532397369327489714492059

comment: $\Delta(4,5,5)$. The minimal polynomial of $\tau$ is $x^{8} - x^{6} - 2 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(4, 5, 6)]:

1.859830244333303664771174558941756842182929403229733768098828266556082268719772546588077549737508729

comment: $\Delta(4,5,6)$. The minimal polynomial of $\tau$ is $x^{12} - 2 x^{9} - 3 x^{8} - 5 x^{7} - 5 x^{6} - 5 x^{5} - 3 x^{4} - 2 x^{3} + 1$.

[(4, 5, 7)]:

1.872104127999394462820878271398550065692670341195048360268486503077900892468315996393290771703517554

comment: $\Delta(4,5,7)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - x^{9} - 3 x^{8} - 4 x^{7} - 3 x^{6} - 4 x^{5} - 3 x^{4} - x^{3} - x^{2} + 1$.

[(4, 5, 8)]:

1.878322686490812265528664459603743258911414291257892683612010905514756227252447487776900155435177873

comment: $\Delta(4,5,8)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - 2 x^{9} - 2 x^{8} - 3 x^{7} - 3 x^{6} - 3 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(4, 5, 9)]:

1.881530431059499083782056060770846366574352520018578881162599146587798316383714982930464499819162928

comment: $\Delta(4,5,9)$. The minimal polynomial of $\tau$ is $x^{16} + x^{15} - 2 x^{13} - 5 x^{12} - 8 x^{11} - 10 x^{10} - 11 x^{9} - 11 x^{8} - 11 x^{7} - 10 x^{6} - 8 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(4, 5, 10)]:

1.883203505913525864168947465362055090560951328672239179570777921570516298917816713755493486651018843

comment: $\Delta(4,5,10)$. The minimal polynomial of $\tau$ is $x^{4} - 2 x^{3} + x^{2} - 2 x + 1$.

[(4, 5, 11)]:

1.884082076596635903789554797540479484860210070209324917467756304093131581882937689156175429479877056

comment: $\Delta(4,5,11)$. The minimal polynomial of $\tau$ is $x^{18} + x^{17} - 2 x^{15} - 5 x^{14} - 8 x^{13} - 10 x^{12} - 11 x^{11} - 11 x^{10} - 11 x^{9} - 11 x^{8} - 11 x^{7} - 10 x^{6} - 8 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(4, 5, 12)]:

1.884545340499252290089150743287297635035297169930261525990223751993224582113974768266406030909169864

comment: $\Delta(4,5,12)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - 2 x^{13} - 2 x^{12} - 3 x^{11} - 3 x^{10} - 3 x^{9} - 2 x^{8} - 3 x^{7} - 3 x^{6} - 3 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(4, 6, 6)]:

1.883203505913525864168947465362055090560951328672239179570777921570516298917816713755493486651018843

comment: $\Delta(4,6,6)$. The minimal polynomial of $\tau$ is $x^{4} - 2 x^{3} + x^{2} - 2 x + 1$.

[(4, 6, 7)]:

1.894633329967184329891735959439529722396815268091426882524452412137772109604614161656140628685183082

comment: $\Delta(4,6,7)$. The minimal polynomial of $\tau$ is $x^{14} - 2 x^{11} - 3 x^{10} - 5 x^{9} - 6 x^{8} - 7 x^{7} - 6 x^{6} - 5 x^{5} - 3 x^{4} - 2 x^{3} + 1$.

[(4, 6, 8)]:

1.900378818101498594844952143137075830605080310231856606823613634347411068677120818748418208246904812

comment: $\Delta(4,6,8)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - 2 x^{9} - 3 x^{7} - x^{6} - 3 x^{5} - 2 x^{3} - x + 1$.

[(4, 6, 9)]:

1.903315999310119930964506103706920323845448819993344579313693415894534011794505515641544155875241441

comment: $\Delta(4,6,9)$. The minimal polynomial of $\tau$ is $x^{14} - x^{13} - x^{11} - 2 x^{10} - 2 x^{9} - 2 x^{8} - 3 x^{7} - 2 x^{6} - 2 x^{5} - 2 x^{4} - x^{3} - x + 1$.

[(4, 6, 10)]:

1.904833083734722820627839954382592669201992200252677351849169691420566064070124560364777187095336486

comment: $\Delta(4,6,10)$. The minimal polynomial of $\tau$ is $x^{16} - x^{15} + x^{14} - 3 x^{13} - 5 x^{11} - x^{10} - 6 x^{9} - x^{8} - 6 x^{7} - x^{6} - 5 x^{5} - 3 x^{3} + x^{2} - x + 1$.

[(4, 6, 11)]:

1.905621594715332556045136947085141480463485825984255753203916703530324178396691411781110240492025086

comment: $\Delta(4,6,11)$. The minimal polynomial of $\tau$ is $x^{10} - x^{8} - 2 x^{7} - 3 x^{6} - 3 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(4, 6, 12)]:

1.906032973382261698104956200492744402492820964633600616531983292817253361457559327715732002018284469

comment: $\Delta(4,6,12)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} - x^{9} - x^{7} - x^{5} - x^{3} - x^{2} - x + 1$.

[(4, 7, 7)]:

1.905621594715332556045136947085141480463485825984255753203916703530324178396691411781110240492025086

comment: $\Delta(4,7,7)$. The minimal polynomial of $\tau$ is $x^{10} - x^{8} - 2 x^{7} - 3 x^{6} - 3 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(4, 7, 8)]:

1.911125065592280900763756117772792433757555181301012059664922525412105343267677218176199969159430361

comment: $\Delta(4,7,8)$. The minimal polynomial of $\tau$ is $x^{10} - 2 x^{8} - 2 x^{7} - x^{6} - x^{5} - x^{4} - 2 x^{3} - 2 x^{2} + 1$.

[(4, 7, 9)]:

1.913926541242641157118865199018127384762704084669186281913360119612695871070354689879607106702008464

comment: $\Delta(4,7,9)$. The minimal polynomial of $\tau$ is $x^{18} + x^{17} - 2 x^{15} - 5 x^{14} - 8 x^{13} - 11 x^{12} - 14 x^{11} - 16 x^{10} - 17 x^{9} - 16 x^{8} - 14 x^{7} - 11 x^{6} - 8 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(4, 7, 10)]:

1.915366775876726534717807507173619641131820605962686396735298073252968724566546999642922587522470370

comment: $\Delta(4,7,10)$. The minimal polynomial of $\tau$ is $x^{18} - 2 x^{15} - 3 x^{14} - 5 x^{13} - 6 x^{12} - 8 x^{11} - 8 x^{10} - 9 x^{9} - 8 x^{8} - 8 x^{7} - 6 x^{6} - 5 x^{5} - 3 x^{4} - 2 x^{3} + 1$.

[(4, 7, 11)]:

1.916111644526957699447533127628994069161896085140814298985502376153221159259028637366200962095936188

comment: $\Delta(4,7,11)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - x^{15} - 3 x^{14} - 4 x^{13} - 4 x^{12} - 6 x^{11} - 6 x^{10} - 5 x^{9} - 6 x^{8} - 6 x^{7} - 4 x^{6} - 4 x^{5} - 3 x^{4} - x^{3} - x^{2} + 1$.

[(4, 7, 12)]:

1.916498263670852969988149642478843213637826266778110447690844606755453077830533665316150905848230937

comment: $\Delta(4,7,12)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} - x^{6} - x^{5} - x^{3} - x^{2} - x + 1$.

[(4, 8, 8)]:

1.916498263670852969988149642478843213637826266778110447690844606755453077830533665316150905848230937

comment: $\Delta(4,8,8)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} - x^{6} - x^{5} - x^{3} - x^{2} - x + 1$.

[(4, 8, 9)]:

1.919227798823979438899995964527130191359788917179917034819870763505834396450951975552715734930979347

comment: $\Delta(4,8,9)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - 2 x^{13} - 2 x^{12} - 3 x^{11} - 4 x^{10} - 5 x^{9} - 5 x^{8} - 5 x^{7} - 4 x^{6} - 3 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(4, 8, 10)]:

1.920627831638399230125177270180970716144784120409648601242960021835967826741501638684151726002435675

comment: $\Delta(4,8,10)$. The minimal polynomial of $\tau$ is $x^{8} - 3 x^{7} + 3 x^{6} - 2 x^{5} + x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 1$.

[(4, 8, 11)]:

1.921350138309322877873548731985072958915566913888236948966457529508687453715899784889903062718904168

comment: $\Delta(4,8,11)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - 2 x^{15} - 2 x^{14} - 3 x^{13} - 4 x^{12} - 5 x^{11} - 5 x^{10} - 5 x^{9} - 5 x^{8} - 5 x^{7} - 4 x^{6} - 3 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(4, 8, 12)]:

1.921724093389319558517536111595229530614354916803858982318556708361099011282810915350419810860190957

comment: $\Delta(4,8,12)$. The minimal polynomial of $\tau$ is $x^{16} - x^{15} - x^{14} - x^{13} + x^{12} - 2 x^{11} - 2 x^{10} - 2 x^{9} + x^{8} - 2 x^{7} - 2 x^{6} - 2 x^{5} + x^{4} - x^{3} - x^{2} - x + 1$.

[(4, 9, 9)]:

1.921918098048184022563130385041329311024704531269774341625563373683280787814634944443835171251438385

comment: $\Delta(4,9,9)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - 2 x^{9} - 3 x^{8} - 3 x^{7} - 3 x^{6} - 3 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(4, 9, 10)]:

1.923296434975780807227210982117206599832886143612779158881465474691513544770860180244551624349941500

comment: $\Delta(4,9,10)$. The minimal polynomial of $\tau$ is $x^{20} - 2 x^{17} - 3 x^{16} - 5 x^{15} - 6 x^{14} - 8 x^{13} - 9 x^{12} - 11 x^{11} - 11 x^{10} - 11 x^{9} - 9 x^{8} - 8 x^{7} - 6 x^{6} - 5 x^{5} - 3 x^{4} - 2 x^{3} + 1$.

[(4, 9, 11)]:

1.924006676489800202497037642539285811669286244790910834483794493070448387620468882990820343539064999

comment: $\Delta(4,9,11)$. The minimal polynomial of $\tau$ is $x^{22} + x^{21} - 2 x^{19} - 5 x^{18} - 8 x^{17} - 11 x^{16} - 14 x^{15} - 17 x^{14} - 20 x^{13} - 22 x^{12} - 23 x^{11} - 22 x^{10} - 20 x^{9} - 17 x^{8} - 14 x^{7} - 11 x^{6} - 8 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(4, 9, 12)]:

1.924373913994191507693242152004808671870876333927253128648845716542841732264388188142406241641229227

comment: $\Delta(4,9,12)$. The minimal polynomial of $\tau$ is $x^{14} - x^{13} - x^{12} - 2 x^{10} - x^{9} - 2 x^{7} - x^{5} - 2 x^{4} - x^{2} - x + 1$.

[(4, 10, 10)]:

1.924662879404263693832622900941968861260426985518231120721349400798983986918345279094323972783618147

comment: $\Delta(4,10,10)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - 2 x^{9} - x^{8} - 2 x^{7} - x^{6} - 2 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(4, 10, 11)]:

1.925366556698908269056840615142529392616079522254611456353305347013466738406280833268955766727596894

comment: $\Delta(4,10,11)$. The minimal polynomial of $\tau$ is $x^{20} - x^{19} - x^{17} - 2 x^{16} - 2 x^{15} - 2 x^{14} - 4 x^{13} - 3 x^{12} - 4 x^{11} - 5 x^{10} - 4 x^{9} - 3 x^{8} - 4 x^{7} - 2 x^{6} - 2 x^{5} - 2 x^{4} - x^{3} - x + 1$.

[(4, 10, 12)]:

1.925730163165916104709086960394754515877517870207682645827957245942287022813860653867261335591820407

comment: $\Delta(4,10,12)$. The minimal polynomial of $\tau$ is $x^{20} - x^{19} - 2 x^{17} - 3 x^{15} - x^{14} - 4 x^{13} - x^{12} - 5 x^{11} - 2 x^{10} - 5 x^{9} - x^{8} - 4 x^{7} - x^{6} - 3 x^{5} - 2 x^{3} - x + 1$.

[(4, 11, 11)]:

1.926066633605494346990243049839457298473949007345162595021081988745160544584811103612471404458765990

comment: $\Delta(4,11,11)$. The minimal polynomial of $\tau$ is $x^{10} - x^{9} - x^{8} - x^{7} - x^{6} + x^{5} - x^{4} - x^{3} - x^{2} - x + 1$.

[(4, 11, 12)]:

1.926428259357260720727448328673662199215066156228356343537847521224831605192150384860526996584138405

comment: $\Delta(4,11,12)$. The minimal polynomial of $\tau$ is $x^{22} - x^{20} - 2 x^{19} - 2 x^{18} - 3 x^{17} - 4 x^{16} - 5 x^{15} - 5 x^{14} - 6 x^{13} - 7 x^{12} - 8 x^{11} - 7 x^{10} - 6 x^{9} - 5 x^{8} - 5 x^{7} - 4 x^{6} - 3 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(4, 12, 12)]:

1.926788800560880892046304635296478834542092148028121060343127956298001185824287999846595473255465399

comment: $\Delta(4,12,12)$. The minimal polynomial of $\tau$ is $x^{8} - 2 x^{6} - 2 x^{5} - x^{4} - 2 x^{3} - 2 x^{2} + 1$.

[(5, 5, 5)]:

1.883203505913525864168947465362055090560951328672239179570777921570516298917816713755493486651018843

comment: $\Delta(5,5,5)$. The minimal polynomial of $\tau$ is $x^{4} - 2 x^{3} + x^{2} - 2 x + 1$.

[(5, 5, 6)]:

1.905621594715332556045136947085141480463485825984255753203916703530324178396691411781110240492025086

comment: $\Delta(5,5,6)$. The minimal polynomial of $\tau$ is $x^{10} - x^{8} - 2 x^{7} - 3 x^{6} - 3 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(5, 5, 7)]:

1.916498263670852969988149642478843213637826266778110447690844606755453077830533665316150905848230937

comment: $\Delta(5,5,7)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} - x^{6} - x^{5} - x^{3} - x^{2} - x + 1$.

[(5, 5, 8)]:

1.921918098048184022563130385041329311024704531269774341625563373683280787814634944443835171251438385

comment: $\Delta(5,5,8)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - 2 x^{9} - 3 x^{8} - 3 x^{7} - 3 x^{6} - 3 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(5, 5, 9)]:

1.924662879404263693832622900941968861260426985518231120721349400798983986918345279094323972783618147

comment: $\Delta(5,5,9)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - 2 x^{9} - x^{8} - 2 x^{7} - x^{6} - 2 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(5, 5, 10)]:

1.926066633605494346990243049839457298473949007345162595021081988745160544584811103612471404458765990

comment: $\Delta(5,5,10)$. The minimal polynomial of $\tau$ is $x^{10} - x^{9} - x^{8} - x^{7} - x^{6} + x^{5} - x^{4} - x^{3} - x^{2} - x + 1$.

[(5, 5, 11)]:

1.926788800560880892046304635296478834542092148028121060343127956298001185824287999846595473255465399

comment: $\Delta(5,5,11)$. The minimal polynomial of $\tau$ is $x^{8} - 2 x^{6} - 2 x^{5} - x^{4} - 2 x^{3} - 2 x^{2} + 1$.

[(5, 5, 12)]:

1.927161629914966413750062655377748096239941292593692838734199886692461796776358005467986219685848816

comment: $\Delta(5,5,12)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - 2 x^{13} - 3 x^{12} - 3 x^{11} - 3 x^{10} - 3 x^{9} - 3 x^{8} - 3 x^{7} - 3 x^{6} - 3 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(5, 6, 6)]:

1.926788800560880892046304635296478834542092148028121060343127956298001185824287999846595473255465399

comment: $\Delta(5,6,6)$. The minimal polynomial of $\tau$ is $x^{8} - 2 x^{6} - 2 x^{5} - x^{4} - 2 x^{3} - 2 x^{2} + 1$.

[(5, 6, 7)]:

1.936996605237438999177053178554403144257448404697528622331429272021345012583034078166859705536666747

comment: $\Delta(5,6,7)$. The minimal polynomial of $\tau$ is $x^{16} + x^{15} - 2 x^{13} - 5 x^{12} - 9 x^{11} - 13 x^{10} - 16 x^{9} - 17 x^{8} - 16 x^{7} - 13 x^{6} - 9 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(5, 6, 8)]:

1.942045445721813234338726947593724874552921568727124751285691799583591586217899531847245757242506014

comment: $\Delta(5,6,8)$. The minimal polynomial of $\tau$ is $x^{16} - 2 x^{13} - 3 x^{12} - 6 x^{11} - 7 x^{10} - 9 x^{9} - 9 x^{8} - 9 x^{7} - 7 x^{6} - 6 x^{5} - 3 x^{4} - 2 x^{3} + 1$.

[(5, 6, 9)]:

1.944581093803699723991385593871738707767417298173778423128144458073256586388032771028968906342961972

comment: $\Delta(5,6,9)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - x^{13} - 3 x^{12} - 5 x^{11} - 5 x^{10} - 6 x^{9} - 7 x^{8} - 6 x^{7} - 5 x^{6} - 5 x^{5} - 3 x^{4} - x^{3} - x^{2} + 1$.

[(5, 6, 10)]:

1.945866326225777486550931225157108436537910770262659372013145615509218036366325810242101504695314693

comment: $\Delta(5,6,10)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} - x^{9} - x^{7} - 2 x^{6} - x^{5} - x^{3} - x^{2} - x + 1$.

[(5, 6, 11)]:

1.946521350371617151481811086885793950108249170578884341385609580095876302735831187285349100534476587

comment: $\Delta(5,6,11)$. The minimal polynomial of $\tau$ is $x^{20} + x^{19} - 2 x^{17} - 5 x^{16} - 9 x^{15} - 13 x^{14} - 16 x^{13} - 18 x^{12} - 19 x^{11} - 19 x^{10} - 19 x^{9} - 18 x^{8} - 16 x^{7} - 13 x^{6} - 9 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(5, 6, 12)]:

1.946856268271884645292791386475331895424656039160279191207151646939207887264527606124413842825554669

comment: $\Delta(5,6,12)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x + 1$.

[(5, 7, 7)]:

1.946856268271884645292791386475331895424656039160279191207151646939207887264527606124413842825554669

comment: $\Delta(5,7,7)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x + 1$.

[(5, 7, 8)]:

1.951716332475569223049074915330666832179184607109892189885941444863270945203000597033766834208480136

comment: $\Delta(5,7,8)$. The minimal polynomial of $\tau$ is $x^{18} + x^{17} - 2 x^{15} - 5 x^{14} - 9 x^{13} - 13 x^{12} - 17 x^{11} - 20 x^{10} - 21 x^{9} - 20 x^{8} - 17 x^{7} - 13 x^{6} - 9 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(5, 7, 9)]:

1.954147640610222263827960386433216597977056895701855361569444291887089807745314411489769041204063287

comment: $\Delta(5,7,9)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - 2 x^{13} - 2 x^{12} - 4 x^{11} - 5 x^{10} - 6 x^{9} - 5 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(5, 7, 10)]:

1.955374770272982474603633482935508211450760808578257545184283485726841629058034136521609394982063655

comment: $\Delta(5,7,10)$. The minimal polynomial of $\tau$ is $x^{14} - x^{13} - x^{12} - 2 x^{10} - x^{9} - x^{8} - 3 x^{7} - x^{6} - x^{5} - 2 x^{4} - x^{2} - x + 1$.

[(5, 7, 11)]:

1.955997400406360245657018622360311550333068875457307370920853261840502411990501938031020032529167727

comment: $\Delta(5,7,11)$. The minimal polynomial of $\tau$ is $x^{20} - 2 x^{17} - 3 x^{16} - 6 x^{15} - 7 x^{14} - 10 x^{13} - 10 x^{12} - 12 x^{11} - 11 x^{10} - 12 x^{9} - 10 x^{8} - 10 x^{7} - 7 x^{6} - 6 x^{5} - 3 x^{4} - 2 x^{3} + 1$.

[(5, 7, 12)]:

1.956314294555582655948000291240178274459781781069506649627935209989379443123885983598897659515621809

comment: $\Delta(5,7,12)$. The minimal polynomial of $\tau$ is $x^{22} + x^{21} - 2 x^{19} - 5 x^{18} - 9 x^{17} - 13 x^{16} - 17 x^{15} - 20 x^{14} - 22 x^{13} - 23 x^{12} - 23 x^{11} - 23 x^{10} - 22 x^{9} - 20 x^{8} - 17 x^{7} - 13 x^{6} - 9 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(5, 8, 8)]:

1.956475871180681908709545885351785856954559538134650331390292238036048815819314498308285829174121339

comment: $\Delta(5,8,8)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - 2 x^{9} - 3 x^{8} - 4 x^{7} - 4 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(5, 8, 9)]:

1.958852451186303052083500286009153609913541382939242727390733561580360960959684423931095660508834879

comment: $\Delta(5,8,9)$. The minimal polynomial of $\tau$ is $x^{14} - x^{12} - 2 x^{11} - 3 x^{10} - 4 x^{9} - 4 x^{8} - 3 x^{7} - 4 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(5, 8, 10)]:

1.960049499559541424937044640082775050303365196289629529910631762065016043706136325749244890242153121

comment: $\Delta(5,8,10)$. The minimal polynomial of $\tau$ is $x^{16} - x^{15} - 2 x^{13} - x^{12} - 2 x^{11} - 2 x^{10} - 3 x^{9} - 3 x^{8} - 3 x^{7} - 2 x^{6} - 2 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(5, 8, 11)]:

1.960655549970022889916247435778790419649651196579942798875441410570325227233815882204466518640234699

comment: $\Delta(5,8,11)$. The minimal polynomial of $\tau$ is $x^{22} + x^{21} - 2 x^{19} - 5 x^{18} - 9 x^{17} - 13 x^{16} - 17 x^{15} - 21 x^{14} - 24 x^{13} - 26 x^{12} - 27 x^{11} - 26 x^{10} - 24 x^{9} - 21 x^{8} - 17 x^{7} - 13 x^{6} - 9 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(5, 8, 12)]:

1.960963313728632042055239556334693052088162610913797534644717203306958370364107111035913896149772611

comment: $\Delta(5,8,12)$. The minimal polynomial of $\tau$ is $x^{20} - x^{18} - 2 x^{17} - 2 x^{16} - 4 x^{15} - 5 x^{14} - 6 x^{13} - 6 x^{12} - 7 x^{11} - 7 x^{10} - 7 x^{9} - 6 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(5, 9, 9)]:

1.961199599539713723099258208993741767184677667442723912295872384609624374146168059583624232079007520

comment: $\Delta(5,9,9)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - 2 x^{9} - x^{8} - 3 x^{7} - x^{6} - 3 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(5, 9, 10)]:

1.962380637102326777199987050609638212765512106135722350050864544633216458155137946695109800879931796

comment: $\Delta(5,9,10)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - 2 x^{15} - 3 x^{14} - 3 x^{13} - 4 x^{12} - 5 x^{11} - 6 x^{10} - 7 x^{9} - 6 x^{8} - 5 x^{7} - 4 x^{6} - 3 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(5, 9, 11)]:

1.962977941304747118360896737225832536887902105175749698564943612056651040650853899727218063632866054

comment: $\Delta(5,9,11)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - 2 x^{13} - 3 x^{12} - 4 x^{11} - 4 x^{10} - 4 x^{9} - 3 x^{8} - 4 x^{7} - 4 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(5, 9, 12)]:

1.963280926201707494114855272199342997959753490720203804291561914386155613542816266801977561367094632

comment: $\Delta(5,9,12)$. The minimal polynomial of $\tau$ is $x^{22} - x^{20} - x^{19} - 3 x^{18} - 5 x^{17} - 5 x^{16} - 7 x^{15} - 9 x^{14} - 9 x^{13} - 10 x^{12} - 11 x^{11} - 10 x^{10} - 9 x^{9} - 9 x^{8} - 7 x^{7} - 5 x^{6} - 5 x^{5} - 3 x^{4} - x^{3} - x^{2} + 1$.

[(5, 10, 10)]:

1.963553038988824614086472487622223205507961832552939411195893564457994189092750348189694023089439559

comment: $\Delta(5,10,10)$. The minimal polynomial of $\tau$ is $x^{6} - 2 x^{5} - x^{4} + 3 x^{3} - x^{2} - 2 x + 1$.

[(5, 10, 11)]:

1.964145660073269597487544779762576706649802236033637917916384713475593529684498400673060133790854450

comment: $\Delta(5,10,11)$. The minimal polynomial of $\tau$ is $x^{20} - x^{18} - 2 x^{17} - 3 x^{16} - 3 x^{15} - 4 x^{14} - 5 x^{13} - 6 x^{12} - 7 x^{11} - 7 x^{10} - 7 x^{9} - 6 x^{8} - 5 x^{7} - 4 x^{6} - 3 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(5, 10, 12)]:

1.964446102336823631384608304636956744680436674155737845859427403982307286342809957275425039556821210

comment: $\Delta(5,10,12)$. The minimal polynomial of $\tau$ is $x^{20} - x^{19} - 2 x^{17} - x^{16} - 2 x^{15} - 2 x^{14} - 3 x^{13} - 3 x^{12} - 4 x^{11} - 3 x^{10} - 4 x^{9} - 3 x^{8} - 3 x^{7} - 2 x^{6} - 2 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(5, 11, 11)]:

1.964735757124309663807153517737293054408530262721198130280188107076117779687158973243880225065850924

comment: $\Delta(5,11,11)$. The minimal polynomial of $\tau$ is $x^{14} - x^{13} - 2 x^{11} - x^{10} - 3 x^{9} - x^{8} - 3 x^{7} - x^{6} - 3 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(5, 11, 12)]:

1.965034836291837408109926237413130461917256761923643578067161937946308583911515534224752476006802821

comment: $\Delta(5,11,12)$. The minimal polynomial of $\tau$ is $x^{26} + x^{25} - 2 x^{23} - 5 x^{22} - 9 x^{21} - 13 x^{20} - 17 x^{19} - 21 x^{18} - 25 x^{17} - 29 x^{16} - 33 x^{15} - 36 x^{14} - 37 x^{13} - 36 x^{12} - 33 x^{11} - 29 x^{10} - 25 x^{9} - 21 x^{8} - 17 x^{7} - 13 x^{6} - 9 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(5, 12, 12)]:

1.965333182704318786119759312008256720615144971957229329023310713141267771194629252533092307869624997

comment: $\Delta(5,12,12)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - 2 x^{13} - 3 x^{12} - 4 x^{11} - 4 x^{10} - 4 x^{9} - 4 x^{8} - 4 x^{7} - 4 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(6, 6, 6)]:

1.946856268271884645292791386475331895424656039160279191207151646939207887264527606124413842825554669

comment: $\Delta(6,6,6)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x + 1$.

[(6, 6, 7)]:

1.956475871180681908709545885351785856954559538134650331390292238036048815819314498308285829174121339

comment: $\Delta(6,6,7)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - 2 x^{9} - 3 x^{8} - 4 x^{7} - 4 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(6, 6, 8)]:

1.961199599539713723099258208993741767184677667442723912295872384609624374146168059583624232079007520

comment: $\Delta(6,6,8)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - 2 x^{9} - x^{8} - 3 x^{7} - x^{6} - 3 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(6, 6, 9)]:

1.963553038988824614086472487622223205507961832552939411195893564457994189092750348189694023089439559

comment: $\Delta(6,6,9)$. The minimal polynomial of $\tau$ is $x^{6} - 2 x^{5} - x^{4} + 3 x^{3} - x^{2} - 2 x + 1$.

[(6, 6, 10)]:

1.964735757124309663807153517737293054408530262721198130280188107076117779687158973243880225065850924

comment: $\Delta(6,6,10)$. The minimal polynomial of $\tau$ is $x^{14} - x^{13} - 2 x^{11} - x^{10} - 3 x^{9} - x^{8} - 3 x^{7} - x^{6} - 3 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(6, 6, 11)]:

1.965333182704318786119759312008256720615144971957229329023310713141267771194629252533092307869624997

comment: $\Delta(6,6,11)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - 2 x^{13} - 3 x^{12} - 4 x^{11} - 4 x^{10} - 4 x^{9} - 4 x^{8} - 4 x^{7} - 4 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(6, 6, 12)]:

1.965635864646311676934260549298704426575200838505070079576539872244122068022572815840623866476565772

comment: $\Delta(6,6,12)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} + x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x + 1$.

[(6, 7, 7)]:

1.965789481934070755687020874946254197485928954882364750480817279365435296642398791720637346241261280

comment: $\Delta(6,7,7)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - 2 x^{9} - 3 x^{8} - 4 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(6, 7, 8)]:

1.970347927046326418170122720333695809156520453036283111061489280687005947646640033307501462225512052

comment: $\Delta(6,7,8)$. The minimal polynomial of $\tau$ is $x^{14} - x^{13} - 2 x^{11} - x^{10} - 2 x^{9} - 3 x^{8} - 3 x^{7} - 3 x^{6} - 2 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(6, 7, 9)]:

1.972610547672464393917626220205150353051582996514713561320294724981424792233552288859022919211734060

comment: $\Delta(6,7,9)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - x^{15} - 3 x^{14} - 5 x^{13} - 6 x^{12} - 8 x^{11} - 9 x^{10} - 9 x^{9} - 9 x^{8} - 8 x^{7} - 6 x^{6} - 5 x^{5} - 3 x^{4} - x^{3} - x^{2} + 1$.

[(6, 7, 10)]:

1.973743054670117050076701233006785652757455318150594410469481781009292432104347749785189262839776024

comment: $\Delta(6,7,10)$. The minimal polynomial of $\tau$ is $x^{20} - 2 x^{17} - 3 x^{16} - 6 x^{15} - 8 x^{14} - 11 x^{13} - 12 x^{12} - 14 x^{11} - 14 x^{10} - 14 x^{9} - 12 x^{8} - 11 x^{7} - 8 x^{6} - 6 x^{5} - 3 x^{4} - 2 x^{3} + 1$.

[(6, 7, 11)]:

1.974312709980077512251264117000370495887920649192466924152923083385601990822681341391508619410182548

comment: $\Delta(6,7,11)$. The minimal polynomial of $\tau$ is $x^{22} + x^{21} - 2 x^{19} - 5 x^{18} - 9 x^{17} - 14 x^{16} - 19 x^{15} - 23 x^{14} - 26 x^{13} - 28 x^{12} - 29 x^{11} - 28 x^{10} - 26 x^{9} - 23 x^{8} - 19 x^{7} - 14 x^{6} - 9 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(6, 7, 12)]:

1.974600072676210822768303372110629764693811101448038573369619096362126728438734100200391704113963339

comment: $\Delta(6,7,12)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - 2 x^{15} - 3 x^{14} - 4 x^{13} - 4 x^{12} - 5 x^{11} - 5 x^{10} - 5 x^{9} - 5 x^{8} - 5 x^{7} - 4 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(6, 8, 8)]:

1.974818708297705874590082435427965385994651535411266614893446215247539976723723778030588114123756178

comment: $\Delta(6,8,8)$. The minimal polynomial of $\tau$ is $x^{6} - 2 x^{5} + x^{4} - 2 x^{3} + x^{2} - 2 x + 1$.

[(6, 8, 9)]:

1.977033884274631233954994815426196756017147656062466954496024079023122545482070453226738301667656928

comment: $\Delta(6,8,9)$. The minimal polynomial of $\tau$ is $x^{18} - x^{17} - x^{15} - 2 x^{14} - 3 x^{13} - 3 x^{12} - 5 x^{11} - 5 x^{10} - 5 x^{9} - 5 x^{8} - 5 x^{7} - 3 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} - x + 1$.

[(6, 8, 10)]:

1.978140498766997423759996634896168299067811688140507404285900541969541381396471641413107331526062677

comment: $\Delta(6,8,10)$. The minimal polynomial of $\tau$ is $x^{20} - x^{19} + x^{18} - 3 x^{17} - 6 x^{15} - 2 x^{14} - 9 x^{13} - 4 x^{12} - 11 x^{11} - 5 x^{10} - 11 x^{9} - 4 x^{8} - 9 x^{7} - 2 x^{6} - 6 x^{5} - 3 x^{3} + x^{2} - x + 1$.

[(6, 8, 11)]:

1.978695995707068026993268784755072743265488706929825111454857362813605415806985719129842083034310572

comment: $\Delta(6,8,11)$. The minimal polynomial of $\tau$ is $x^{22} - 2 x^{19} - 3 x^{18} - 6 x^{17} - 8 x^{16} - 11 x^{15} - 13 x^{14} - 15 x^{13} - 16 x^{12} - 17 x^{11} - 16 x^{10} - 15 x^{9} - 13 x^{8} - 11 x^{7} - 8 x^{6} - 6 x^{5} - 3 x^{4} - 2 x^{3} + 1$.

[(6, 8, 12)]:

1.978975626890489385880860017674181236617222427546451770013695256795200187590921833011991487282521431

comment: $\Delta(6,8,12)$. The minimal polynomial of $\tau$ is $x^{12} - 2 x^{11} + x^{8} - x^{7} - x^{6} - x^{5} + x^{4} - 2 x + 1$.

[(6, 9, 9)]:

1.979223695577036596226525684695597033498339753727263947927810428801656761501218633193522409982183500

comment: $\Delta(6,9,9)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} - 2 x^{8} - 2 x^{7} - x^{6} - 2 x^{5} - 2 x^{4} - x^{2} - x + 1$.

[(6, 9, 10)]:

1.980316609212873224673286325691617286770118767994977181767125683583828785551728847069527321953072575

comment: $\Delta(6,9,10)$. The minimal polynomial of $\tau$ is $x^{14} - x^{13} - x^{12} - 2 x^{10} - 2 x^{9} - x^{8} - 2 x^{7} - x^{6} - 2 x^{5} - 2 x^{4} - x^{2} - x + 1$.

[(6, 9, 11)]:

1.980864680391380208056132318309254339105689763048538424117487412192204554085502202483028651243133607

comment: $\Delta(6,9,11)$. The minimal polynomial of $\tau$ is $x^{18} - x^{17} - x^{16} - 2 x^{14} - x^{13} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} - 2 x^{8} - 3 x^{7} - 2 x^{6} - x^{5} - 2 x^{4} - x^{2} - x + 1$.

[(6, 9, 12)]:

1.981140287924534323189903987723660832523064416694833529125771060089180729732802115416474095324018356

comment: $\Delta(6,9,12)$. The minimal polynomial of $\tau$ is $x^{18} - x^{17} - x^{16} - 2 x^{14} - 2 x^{13} - 3 x^{11} - 3 x^{10} - x^{9} - 3 x^{8} - 3 x^{7} - 2 x^{5} - 2 x^{4} - x^{2} - x + 1$.

[(6, 10, 10)]:

1.981402184988944055358786372739850697104736748196229205013567703017436584719097755469361679971586641

comment: $\Delta(6,10,10)$. The minimal polynomial of $\tau$ is $x^{14} - x^{13} - 2 x^{11} - x^{10} - 3 x^{9} - 2 x^{8} - 3 x^{7} - 2 x^{6} - 3 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(6, 10, 11)]:

1.981946307818925636529858443095333706763243080308571773749852486253775202651695594487439379646311782

comment: $\Delta(6,10,11)$. The minimal polynomial of $\tau$ is $x^{24} - 2 x^{21} - 3 x^{20} - 6 x^{19} - 8 x^{18} - 11 x^{17} - 13 x^{16} - 16 x^{15} - 18 x^{14} - 20 x^{13} - 20 x^{12} - 20 x^{11} - 18 x^{10} - 16 x^{9} - 13 x^{8} - 11 x^{7} - 8 x^{6} - 6 x^{5} - 3 x^{4} - 2 x^{3} + 1$.

[(6, 10, 12)]:

1.982219789362919888389261196131583755056106831021252191486644234774748159824785486973360907679911350

comment: $\Delta(6,10,12)$. The minimal polynomial of $\tau$ is $x^{14} - x^{13} - 3 x^{11} - 3 x^{9} + x^{8} - 3 x^{7} + x^{6} - 3 x^{5} - 3 x^{3} - x + 1$.

[(6, 11, 11)]:

1.982488319105713147008328676127456400888741746090538128953487054735487432558712665613588198226391178

comment: $\Delta(6,11,11)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - 2 x^{13} - 3 x^{12} - 4 x^{11} - 5 x^{10} - 5 x^{9} - 5 x^{8} - 5 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(6, 11, 12)]:

1.982760669697902440178187657619136699685645409524569899949787204723106686023462185508358060661074899

comment: $\Delta(6,11,12)$. The minimal polynomial of $\tau$ is $x^{22} - x^{20} - 2 x^{19} - 3 x^{18} - 4 x^{17} - 4 x^{16} - 5 x^{15} - 6 x^{14} - 7 x^{13} - 8 x^{12} - 9 x^{11} - 8 x^{10} - 7 x^{9} - 6 x^{8} - 5 x^{7} - 4 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(6, 12, 12)]:

1.983032417311542006114549216723423379705134097500687312332433350792107074562756897374070157163255810

comment: $\Delta(6,12,12)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{5} - x^{4} - x^{3} - x^{2} - x + 1$.

[(7, 7, 7)]:

1.974818708297705874590082435427965385994651535411266614893446215247539976723723778030588114123756178

comment: $\Delta(7,7,7)$. The minimal polynomial of $\tau$ is $x^{6} - 2 x^{5} + x^{4} - 2 x^{3} + x^{2} - 2 x + 1$.

[(7, 7, 8)]:

1.979223695577036596226525684695597033498339753727263947927810428801656761501218633193522409982183500

comment: $\Delta(7,7,8)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} - 2 x^{8} - 2 x^{7} - x^{6} - 2 x^{5} - 2 x^{4} - x^{2} - x + 1$.

[(7, 7, 9)]:

1.981402184988944055358786372739850697104736748196229205013567703017436584719097755469361679971586641

comment: $\Delta(7,7,9)$. The minimal polynomial of $\tau$ is $x^{14} - x^{13} - 2 x^{11} - x^{10} - 3 x^{9} - 2 x^{8} - 3 x^{7} - 2 x^{6} - 3 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(7, 7, 10)]:

1.982488319105713147008328676127456400888741746090538128953487054735487432558712665613588198226391178

comment: $\Delta(7,7,10)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - 2 x^{13} - 3 x^{12} - 4 x^{11} - 5 x^{10} - 5 x^{9} - 5 x^{8} - 5 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(7, 7, 11)]:

1.983032417311542006114549216723423379705134097500687312332433350792107074562756897374070157163255810

comment: $\Delta(7,7,11)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{5} - x^{4} - x^{3} - x^{2} - x + 1$.

[(7, 7, 12)]:

1.983305735750005776375189012628317741868815722882883516189595631824828361197820256011900515804455165

comment: $\Delta(7,7,12)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - 2 x^{15} - 3 x^{14} - 4 x^{13} - 5 x^{12} - 5 x^{11} - 5 x^{10} - 5 x^{9} - 5 x^{8} - 5 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(7, 8, 8)]:

1.983547387915046507708271861503548004199943801308628706334305771304885859810287548882538203623283603

comment: $\Delta(7,8,8)$. The minimal polynomial of $\tau$ is $x^{14} - x^{12} - 2 x^{11} - 3 x^{10} - 4 x^{9} - 5 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(7, 8, 9)]:

1.985681982695484213427515982651103771789564409717833298788726510453612618136190143784614610684204548

comment: $\Delta(7,8,9)$. The minimal polynomial of $\tau$ is $x^{22} + x^{21} - 2 x^{19} - 5 x^{18} - 9 x^{17} - 14 x^{16} - 20 x^{15} - 26 x^{14} - 31 x^{13} - 34 x^{12} - 35 x^{11} - 34 x^{10} - 31 x^{9} - 26 x^{8} - 20 x^{7} - 14 x^{6} - 9 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(7, 8, 10)]:

1.986744234944723412104364958360452128032386972049892349066834585996120582799248151208511008204615571

comment: $\Delta(7,8,10)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - 2 x^{15} - 3 x^{14} - 4 x^{13} - 4 x^{12} - 6 x^{11} - 7 x^{10} - 7 x^{9} - 7 x^{8} - 6 x^{7} - 4 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(7, 8, 11)]:

1.987275320060982381461011782850450560446554526270330637200434459587012373805332772563130710733529436

comment: $\Delta(7,8,11)$. The minimal polynomial of $\tau$ is $x^{20} - x^{18} - 2 x^{17} - 3 x^{16} - 3 x^{15} - 5 x^{14} - 7 x^{13} - 8 x^{12} - 8 x^{11} - 7 x^{10} - 8 x^{9} - 8 x^{8} - 7 x^{7} - 5 x^{6} - 3 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(7, 8, 12)]:

1.987541559379913793798496054730958979164095195880858100034539545406859135624631201175439843471199164

comment: $\Delta(7,8,12)$. The minimal polynomial of $\tau$ is $x^{22} - x^{20} - 2 x^{19} - 2 x^{18} - 4 x^{17} - 6 x^{16} - 8 x^{15} - 8 x^{14} - 9 x^{13} - 10 x^{12} - 11 x^{11} - 10 x^{10} - 9 x^{9} - 8 x^{8} - 8 x^{7} - 6 x^{6} - 4 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} + 1$.

[(7, 9, 9)]:

1.987793166846224434374390262230576844343393924539143017482918571285237516585331264132639554991353175

comment: $\Delta(7,9,9)$. The minimal polynomial of $\tau$ is $x^{6} - 2 x^{4} - 3 x^{3} - 2 x^{2} + 1$.

[(7, 9, 10)]:

1.988842812647093128769963590483093565566089056654043025222990011003795673473586948353821248067138882

comment: $\Delta(7,9,10)$. The minimal polynomial of $\tau$ is $x^{24} + x^{23} - 2 x^{21} - 5 x^{20} - 9 x^{19} - 14 x^{18} - 20 x^{17} - 26 x^{16} - 32 x^{15} - 37 x^{14} - 40 x^{13} - 41 x^{12} - 40 x^{11} - 37 x^{10} - 32 x^{9} - 26 x^{8} - 20 x^{7} - 14 x^{6} - 9 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(7, 9, 11)]:

1.989367089140442447346965705454714173950646877010461365924337896466721570249260431810708808750671969

comment: $\Delta(7,9,11)$. The minimal polynomial of $\tau$ is $x^{24} - 2 x^{21} - 3 x^{20} - 6 x^{19} - 8 x^{18} - 12 x^{17} - 14 x^{16} - 18 x^{15} - 19 x^{14} - 22 x^{13} - 21 x^{12} - 22 x^{11} - 19 x^{10} - 18 x^{9} - 14 x^{8} - 12 x^{7} - 8 x^{6} - 6 x^{5} - 3 x^{4} - 2 x^{3} + 1$.

[(7, 9, 12)]:

1.989629653158333087758262896832702548371992884403752722988517518704938253607213869008705330096436470

comment: $\Delta(7,9,12)$. The minimal polynomial of $\tau$ is $x^{24} - x^{22} - x^{21} - 3 x^{20} - 5 x^{19} - 6 x^{18} - 9 x^{17} - 11 x^{16} - 12 x^{15} - 14 x^{14} - 15 x^{13} - 15 x^{12} - 15 x^{11} - 14 x^{10} - 12 x^{9} - 11 x^{8} - 9 x^{7} - 6 x^{6} - 5 x^{5} - 3 x^{4} - x^{3} - x^{2} + 1$.

[(7, 10, 10)]:

1.989885727579599072692845278020843010311665626634770842783513846718374947446642571415948524424551572

comment: $\Delta(7,10,10)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - 2 x^{13} - 3 x^{12} - 4 x^{11} - 5 x^{10} - 6 x^{9} - 6 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(7, 10, 11)]:

1.990406395050066277160288544205431883854632078093577335575429875758751837693899859504949316625788971

comment: $\Delta(7,10,11)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - 2 x^{15} - 3 x^{14} - 4 x^{13} - 5 x^{12} - 6 x^{11} - 6 x^{10} - 5 x^{9} - 6 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(7, 10, 12)]:

1.990667023127863227531848771239637266275464424712627186748291073395704221540075386323130344708340548

comment: $\Delta(7,10,12)$. The minimal polynomial of $\tau$ is $x^{26} - 2 x^{23} - 3 x^{22} - 6 x^{21} - 8 x^{20} - 12 x^{19} - 14 x^{18} - 18 x^{17} - 20 x^{16} - 23 x^{15} - 24 x^{14} - 25 x^{13} - 24 x^{12} - 23 x^{11} - 20 x^{10} - 18 x^{9} - 14 x^{8} - 12 x^{7} - 8 x^{6} - 6 x^{5} - 3 x^{4} - 2 x^{3} + 1$.

[(7, 11, 11)]:

1.990925139223829841504083273674656436242002214360552001457338650035707030868595903505534722471263514

comment: $\Delta(7,11,11)$. The minimal polynomial of $\tau$ is $x^{16} - x^{15} - 2 x^{13} - x^{12} - 3 x^{11} - 2 x^{10} - 4 x^{9} - 2 x^{8} - 4 x^{7} - 2 x^{6} - 3 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(7, 11, 12)]:

1.991184741064290783710021017750894002020557818883959205131636728310177618988921141006186570640073982

comment: $\Delta(7,11,12)$. The minimal polynomial of $\tau$ is $x^{28} + x^{27} - 2 x^{25} - 5 x^{24} - 9 x^{23} - 14 x^{22} - 20 x^{21} - 26 x^{20} - 32 x^{19} - 38 x^{18} - 44 x^{17} - 49 x^{16} - 52 x^{15} - 53 x^{14} - 52 x^{13} - 49 x^{12} - 44 x^{11} - 38 x^{10} - 32 x^{9} - 26 x^{8} - 20 x^{7} - 14 x^{6} - 9 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(7, 12, 12)]:

1.991443797820922287578110199505070672293533618274102414102698776172948394787166947350900263477535520

comment: $\Delta(7,12,12)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - 2 x^{15} - 3 x^{14} - 4 x^{13} - 5 x^{12} - 6 x^{11} - 6 x^{10} - 6 x^{9} - 6 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(8, 8, 8)]:

1.987793166846224434374390262230576844343393924539143017482918571285237516585331264132639554991353175

comment: $\Delta(8,8,8)$. The minimal polynomial of $\tau$ is $x^{6} - 2 x^{4} - 3 x^{3} - 2 x^{2} + 1$.

[(8, 8, 9)]:

1.989885727579599072692845278020843010311665626634770842783513846718374947446642571415948524424551572

comment: $\Delta(8,8,9)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - 2 x^{13} - 3 x^{12} - 4 x^{11} - 5 x^{10} - 6 x^{9} - 6 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(8, 8, 10)]:

1.990925139223829841504083273674656436242002214360552001457338650035707030868595903505534722471263514

comment: $\Delta(8,8,10)$. The minimal polynomial of $\tau$ is $x^{16} - x^{15} - 2 x^{13} - x^{12} - 3 x^{11} - 2 x^{10} - 4 x^{9} - 2 x^{8} - 4 x^{7} - 2 x^{6} - 3 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(8, 8, 11)]:

1.991443797820922287578110199505070672293533618274102414102698776172948394787166947350900263477535520

comment: $\Delta(8,8,11)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - 2 x^{15} - 3 x^{14} - 4 x^{13} - 5 x^{12} - 6 x^{11} - 6 x^{10} - 6 x^{9} - 6 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(8, 8, 12)]:

1.991703288721262216525852404851603217748145270213722041930314850989403677469578346865696584225421855

comment: $\Delta(8,8,12)$. The minimal polynomial of $\tau$ is $x^{16} - x^{15} - x^{14} - x^{13} - 2 x^{11} - 2 x^{10} - 2 x^{9} - 2 x^{7} - 2 x^{6} - 2 x^{5} - x^{3} - x^{2} - x + 1$.

[(8, 9, 9)]:

1.991955889957806501452044760315755614897019043126873961493286097194499270725558292081423448835619863

comment: $\Delta(8,9,9)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - 2 x^{13} - 3 x^{12} - 4 x^{11} - 5 x^{10} - 6 x^{9} - 7 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(8, 9, 10)]:

1.992983255999099547884598196927875149809866610796472321141152016856151123520682882477704039838515405

comment: $\Delta(8,9,10)$. The minimal polynomial of $\tau$ is $x^{24} - 2 x^{21} - 3 x^{20} - 6 x^{19} - 8 x^{18} - 12 x^{17} - 15 x^{16} - 19 x^{15} - 21 x^{14} - 23 x^{13} - 23 x^{12} - 23 x^{11} - 21 x^{10} - 19 x^{9} - 15 x^{8} - 12 x^{7} - 8 x^{6} - 6 x^{5} - 3 x^{4} - 2 x^{3} + 1$.

[(8, 9, 11)]:

1.993495419048330129476132305176172014651769467385457335134498942189750649710306066248852477213019928

comment: $\Delta(8,9,11)$. The minimal polynomial of $\tau$ is $x^{26} + x^{25} - 2 x^{23} - 5 x^{22} - 9 x^{21} - 14 x^{20} - 20 x^{19} - 27 x^{18} - 34 x^{17} - 40 x^{16} - 45 x^{15} - 48 x^{14} - 49 x^{13} - 48 x^{12} - 45 x^{11} - 40 x^{10} - 34 x^{9} - 27 x^{8} - 20 x^{7} - 14 x^{6} - 9 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(8, 9, 12)]:

1.993751409695060277379368171139662096197335464792700944228190948160677683825316769733195850987529077

comment: $\Delta(8,9,12)$. The minimal polynomial of $\tau$ is $x^{16} - 2 x^{15} + x^{13} - x^{12} - 2 x^{11} + x^{10} - 3 x^{8} + x^{6} - 2 x^{5} - x^{4} + x^{3} - 2 x + 1$.

[(8, 10, 10)]:

1.994004199185753973768510255076832811456043257197981166539933355079807033564438310048874070908904621

comment: $\Delta(8,10,10)$. The minimal polynomial of $\tau$ is $x^{8} - 2 x^{7} + x^{6} - 2 x^{5} + x^{4} - 2 x^{3} + x^{2} - 2 x + 1$.

[(8, 10, 11)]:

1.994512923807972049556299619205595437164597668002399406874503490839130343059873136154993970271337794

comment: $\Delta(8,10,11)$. The minimal polynomial of $\tau$ is $x^{20} - x^{19} - 2 x^{17} - x^{16} - 3 x^{15} - 2 x^{14} - 3 x^{13} - 4 x^{12} - 4 x^{11} - 5 x^{10} - 4 x^{9} - 4 x^{8} - 3 x^{7} - 2 x^{6} - 3 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(8, 10, 12)]:

1.994767073226062558911895692124911699415414946075015409520207157562313183065736495710563775479809596

comment: $\Delta(8,10,12)$. The minimal polynomial of $\tau$ is $x^{24} - x^{23} - 2 x^{21} - 4 x^{19} - 2 x^{18} - 6 x^{17} - 3 x^{16} - 8 x^{15} - 5 x^{14} - 9 x^{13} - 5 x^{12} - 9 x^{11} - 5 x^{10} - 8 x^{9} - 3 x^{8} - 6 x^{7} - 2 x^{6} - 4 x^{5} - 2 x^{3} - x + 1$.

[(8, 11, 11)]:

1.995019818966310460081389463810779645944306794562223016585492179267052420061167866699955706319271927

comment: $\Delta(8,11,11)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - 2 x^{15} - 3 x^{14} - 4 x^{13} - 5 x^{12} - 6 x^{11} - 7 x^{10} - 7 x^{9} - 7 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(8, 11, 12)]:

1.995272993918165302897140035818543400601617641229606165867657758741429612911762264098133004407504807

comment: $\Delta(8,11,12)$. The minimal polynomial of $\tau$ is $x^{18} - 2 x^{16} - 2 x^{15} - x^{14} - 2 x^{13} - 4 x^{12} - 4 x^{11} - 3 x^{10} - 3 x^{9} - 3 x^{8} - 4 x^{7} - 4 x^{6} - 2 x^{5} - x^{4} - 2 x^{3} - 2 x^{2} + 1$.

[(8, 12, 12)]:

1.995525652191791588299921649920726784798985687096781824711337824353127872999804502474647073242565761

comment: $\Delta(8,12,12)$. The minimal polynomial of $\tau$ is $x^{16} - x^{15} - x^{14} - x^{13} - 2 x^{11} - 2 x^{10} - 2 x^{9} - x^{8} - 2 x^{7} - 2 x^{6} - 2 x^{5} - x^{3} - x^{2} - x + 1$.

[(9, 9, 9)]:

1.994004199185753973768510255076832811456043257197981166539933355079807033564438310048874070908904621

comment: $\Delta(9,9,9)$. The minimal polynomial of $\tau$ is $x^{8} - 2 x^{7} + x^{6} - 2 x^{5} + x^{4} - 2 x^{3} + x^{2} - 2 x + 1$.

[(9, 9, 10)]:

1.995019818966310460081389463810779645944306794562223016585492179267052420061167866699955706319271927

comment: $\Delta(9,9,10)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - 2 x^{15} - 3 x^{14} - 4 x^{13} - 5 x^{12} - 6 x^{11} - 7 x^{10} - 7 x^{9} - 7 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(9, 9, 11)]:

1.995525652191791588299921649920726784798985687096781824711337824353127872999804502474647073242565761

comment: $\Delta(9,9,11)$. The minimal polynomial of $\tau$ is $x^{16} - x^{15} - x^{14} - x^{13} - 2 x^{11} - 2 x^{10} - 2 x^{9} - x^{8} - 2 x^{7} - 2 x^{6} - 2 x^{5} - x^{3} - x^{2} - x + 1$.

[(9, 9, 12)]:

1.995778234544805625063722312939359795303375925287441385104211549842281074449348774714189131822180815

comment: $\Delta(9,9,12)$. The minimal polynomial of $\tau$ is $x^{18} - x^{17} - x^{16} - 2 x^{14} - 2 x^{13} - x^{12} - 3 x^{11} - 3 x^{10} - x^{9} - 3 x^{8} - 3 x^{7} - x^{6} - 2 x^{5} - 2 x^{4} - x^{2} - x + 1$.

[(9, 10, 10)]:

1.996029178945384038710233532335590385049169844297504922127827424907139651736780424173111122532029354

comment: $\Delta(9,10,10)$. The minimal polynomial of $\tau$ is $x^{16} - 2 x^{14} - 2 x^{13} - x^{12} - 2 x^{11} - 4 x^{10} - 4 x^{9} - 3 x^{8} - 4 x^{7} - 4 x^{6} - 2 x^{5} - x^{4} - 2 x^{3} - 2 x^{2} + 1$.

[(9, 10, 11)]:

1.996531663375060760771853253229288018856488424843109527132759119710384990147796729194749812883697036

comment: $\Delta(9,10,11)$. The minimal polynomial of $\tau$ is $x^{28} + x^{27} - 2 x^{25} - 5 x^{24} - 9 x^{23} - 14 x^{22} - 20 x^{21} - 27 x^{20} - 35 x^{19} - 43 x^{18} - 50 x^{17} - 55 x^{16} - 58 x^{15} - 59 x^{14} - 58 x^{13} - 55 x^{12} - 50 x^{11} - 43 x^{10} - 35 x^{9} - 27 x^{8} - 20 x^{7} - 14 x^{6} - 9 x^{5} - 5 x^{4} - 2 x^{3} + x + 1$.

[(9, 10, 12)]:

1.996782453916014191924714207361339968000636630943980149282260714781835024822426997081745948107076631

comment: $\Delta(9,10,12)$. The minimal polynomial of $\tau$ is $x^{26} - x^{25} - x^{23} - 2 x^{22} - 3 x^{21} - 3 x^{20} - 6 x^{19} - 6 x^{18} - 8 x^{17} - 9 x^{16} - 10 x^{15} - 10 x^{14} - 11 x^{13} - 10 x^{12} - 10 x^{11} - 9 x^{10} - 8 x^{9} - 6 x^{8} - 6 x^{7} - 3 x^{6} - 3 x^{5} - 2 x^{4} - x^{3} - x + 1$.

[(9, 11, 11)]:

1.997032367302200037004409570556926340005914281690467212974962820085857410616441543755581373768259082

comment: $\Delta(9,11,11)$. The minimal polynomial of $\tau$ is $x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x + 1$.

[(9, 11, 12)]:

1.997282210218245656462367797119343298735089825953992244095826774191069705491705565163120317962373740

comment: $\Delta(9,11,12)$. The minimal polynomial of $\tau$ is $x^{28} - x^{26} - x^{25} - 3 x^{24} - 5 x^{23} - 6 x^{22} - 9 x^{21} - 12 x^{20} - 14 x^{19} - 17 x^{18} - 20 x^{17} - 21 x^{16} - 22 x^{15} - 23 x^{14} - 22 x^{13} - 21 x^{12} - 20 x^{11} - 17 x^{10} - 14 x^{9} - 12 x^{8} - 9 x^{7} - 6 x^{6} - 5 x^{5} - 3 x^{4} - x^{3} - x^{2} + 1$.

[(9, 12, 12)]:

1.997531551096235425506831561618914300847830408423510720054170797050134566713362599404134180331240131

comment: $\Delta(9,12,12)$. The minimal polynomial of $\tau$ is $x^{14} - x^{13} - x^{12} - 3 x^{10} - x^{9} - 3 x^{7} - x^{5} - 3 x^{4} - x^{2} - x + 1$.

[(10, 10, 10)]:

1.997032367302200037004409570556926340005914281690467212974962820085857410616441543755581373768259082

comment: $\Delta(10,10,10)$. The minimal polynomial of $\tau$ is $x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x + 1$.

[(10, 10, 11)]:

1.997531551096235425506831561618914300847830408423510720054170797050134566713362599404134180331240131

comment: $\Delta(10,10,11)$. The minimal polynomial of $\tau$ is $x^{14} - x^{13} - x^{12} - 3 x^{10} - x^{9} - 3 x^{7} - x^{5} - 3 x^{4} - x^{2} - x + 1$.

[(10, 10, 12)]:

1.997780576221445683213966002508360418740106173195739663671208462952170079220545495849289875059549976

comment: $\Delta(10,10,12)$. The minimal polynomial of $\tau$ is $x^{20} - x^{19} - 2 x^{17} - x^{16} - 3 x^{15} - 2 x^{14} - 4 x^{13} - 3 x^{12} - 5 x^{11} - 3 x^{10} - 5 x^{9} - 3 x^{8} - 4 x^{7} - 2 x^{6} - 3 x^{5} - x^{4} - 2 x^{3} - x + 1$.

[(10, 11, 11)]:

1.998028980540465304018156441516887554188243201267682719421131861974734074199456800099175329275092343

comment: $\Delta(10,11,11)$. The minimal polynomial of $\tau$ is $x^{20} - x^{18} - 2 x^{17} - 3 x^{16} - 4 x^{15} - 5 x^{14} - 6 x^{13} - 7 x^{12} - 8 x^{11} - 9 x^{10} - 8 x^{9} - 7 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(10, 11, 12)]:

1.998277072300828633681009963652519786992791344025189166722054782425194420006460394045958250653762060

comment: $\Delta(10,11,12)$. The minimal polynomial of $\tau$ is $x^{30} - 2 x^{27} - 3 x^{26} - 6 x^{25} - 8 x^{24} - 12 x^{23} - 15 x^{22} - 20 x^{21} - 24 x^{20} - 29 x^{19} - 32 x^{18} - 35 x^{17} - 36 x^{16} - 37 x^{15} - 36 x^{14} - 35 x^{13} - 32 x^{12} - 29 x^{11} - 24 x^{10} - 20 x^{9} - 15 x^{8} - 12 x^{7} - 8 x^{6} - 6 x^{5} - 3 x^{4} - 2 x^{3} + 1$.

[(10, 12, 12)]:

1.998524669762450564155370787352761166092246315063360007852376867750317275895120154807847787215564447

comment: $\Delta(10,12,12)$. The minimal polynomial of $\tau$ is $x^{10} - 2 x^{9} + x^{8} - 2 x^{7} + x^{6} - 2 x^{5} + x^{4} - 2 x^{3} + x^{2} - 2 x + 1$.

[(11, 11, 11)]:

1.998524669762450564155370787352761166092246315063360007852376867750317275895120154807847787215564447

comment: $\Delta(11,11,11)$. The minimal polynomial of $\tau$ is $x^{10} - 2 x^{9} + x^{8} - 2 x^{7} + x^{6} - 2 x^{5} + x^{4} - 2 x^{3} + x^{2} - 2 x + 1$.

[(11, 11, 12)]:

1.998771835824647307945086882412041711024465553163350203369433266461192430251273952427655830932380825

comment: $\Delta(11,11,12)$. The minimal polynomial of $\tau$ is $x^{22} - x^{20} - 2 x^{19} - 3 x^{18} - 4 x^{17} - 5 x^{16} - 6 x^{15} - 7 x^{14} - 8 x^{13} - 9 x^{12} - 9 x^{11} - 9 x^{10} - 8 x^{9} - 7 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(11, 12, 12)]:

1.999018511730854992232175987736832523813130332656057036455879798332349403238212387745278724021636156

comment: $\Delta(11,12,12)$. The minimal polynomial of $\tau$ is $x^{22} - x^{20} - 2 x^{19} - 3 x^{18} - 4 x^{17} - 5 x^{16} - 6 x^{15} - 7 x^{14} - 8 x^{13} - 9 x^{12} - 10 x^{11} - 9 x^{10} - 8 x^{9} - 7 x^{8} - 6 x^{7} - 5 x^{6} - 4 x^{5} - 3 x^{4} - 2 x^{3} - x^{2} + 1$.

[(12, 12, 12)]:

1.999264699710832703686343133931738518235540964719784909507895366112354954903856176121117174743082362

comment: $\Delta(12,12,12)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x + 1$.

[inf, 3]:

1.324717957244746025960908854478097340734404056901733364534015050302827851245547594054699347981787280

comment: $\Delta(2,3,\infty)$. The minimal polynomial of $\tau$ is $x^{3} - x - 1$. This is the plastic number.

[inf, 4]:

1.465571231876768026656731225219939108025577568472285701643183111249262996685017840478125801194909270

comment: $\Delta(2,4,\infty)$. The minimal polynomial of $\tau$ is $x^{3} - x^{2} - 1$.

[inf, 5]:

1.534157744914266915435970076109375701882545038516595135368531863008063023214082281436789664835483494

comment: $\Delta(2,5,\infty)$. The minimal polynomial of $\tau$ is $x^{5} - x^{3} - x^{2} - x - 1$.

[inf, 6]:

1.570147312196054362910665435137126553873131607424527436931654877897330661544162320222760040702620119

comment: $\Delta(2,6,\infty)$. The minimal polynomial of $\tau$ is $x^{5} - x^{4} - x^{2} - 1$.

[inf, 7]:

1.590005373901363925162015541663808196898573270019121975693623393697501137164006258637372010170673034

comment: $\Delta(2,7,\infty)$. The minimal polynomial of $\tau$ is $x^{7} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

[inf, 8]:

1.601347333787636724232618164313101117665668726676556628193381197227260290102478499805029985176068445

comment: $\Delta(2,8,\infty)$. The minimal polynomial of $\tau$ is $x^{7} - x^{6} - x^{4} - x^{2} - 1$.

[inf, 9]:

1.607982727928201149922420457795797515362209291893843383821724387553882342235678008938055023076222152

comment: $\Delta(2,9,\infty)$. The minimal polynomial of $\tau$ is $x^{9} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

[inf, 10]:

1.611930396564119819833599200722692620452821331844328319226299635293730595602174524805609485749220372

comment: $\Delta(2,10,\infty)$. The minimal polynomial of $\tau$ is $x^{9} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

[inf, 11]:

1.614306823257148514569632339911304106948171748039112819042361616990015585866967144509566552856428042

comment: $\Delta(2,11,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

[inf, 12]:

1.615749202755210610743641798789103362281243911263783155770251095174096083205827141199474172669686514

comment: $\Delta(2,12,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

[inf, inf]:

1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137

comment: $\Delta(2,\infty,\infty)$. The minimal polynomial of $\tau$ is $x^{2} - x - 1$. This is the golden ratio.

[(3, 3, inf)]:

1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137

comment: $\Delta(3,3,\infty)$. The minimal polynomial of $\tau$ is $x^{2} - x - 1$. This is the golden ratio.

[(3, 4, inf)]:

1.734691345692469553024170512863266780535219119458219288432296812335205399478460452151703933375357223

comment: $\Delta(3,4,\infty)$. The minimal polynomial of $\tau$ is $x^{5} - x^{3} - 2 x^{2} - 2 x - 1$.

[(3, 5, inf)]:

1.787115545692877030408856544102545477626661555209018932569173737103357878058979257346163073187720897

comment: $\Delta(3,5,\infty)$. The minimal polynomial of $\tau$ is $x^{6} - x^{4} - 2 x^{3} - 2 x^{2} - 2 x - 1$.

[(3, 6, inf)]:

1.812403619268042660789692972549291455045000701472089973357385463329079917206255314773656749402717769

comment: $\Delta(3,6,\infty)$. The minimal polynomial of $\tau$ is $x^{5} - x^{4} - x^{3} - x - 1$.

[(3, 7, inf)]:

1.825151056526864236475139850401598398753850052678498166654493189647011569329025297553430500673727375

comment: $\Delta(3,7,\infty)$. The minimal polynomial of $\tau$ is $x^{8} - x^{6} - 2 x^{5} - 2 x^{4} - 2 x^{3} - 2 x^{2} - 2 x - 1$.

[(3, 8, inf)]:

1.831758982624219313220924392758945909180753284430305528882268041050898033407545012172466310872463610

comment: $\Delta(3,8,\infty)$. The minimal polynomial of $\tau$ is $x^{9} - x^{7} - 2 x^{6} - 2 x^{5} - 2 x^{4} - 2 x^{3} - 2 x^{2} - 2 x - 1$.

[(3, 9, inf)]:

1.835245915136238455139915426637682792426287987073872329242700022447472856310450050991223610038001242

comment: $\Delta(3,9,\infty)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} - x^{6} - x^{4} - x^{3} - x - 1$.

[(3, 10, inf)]:

1.837106874766394548270844321330466648068771338958568913349127084247764399704093547521973424237143361

comment: $\Delta(3,10,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{9} - 2 x^{8} - 2 x^{7} - 2 x^{6} - 2 x^{5} - 2 x^{4} - 2 x^{3} - 2 x^{2} - 2 x - 1$.

[(3, 11, inf)]:

1.838107172876629015587597435469413971525849484095155901197632401024069812055688490964304961247336057

comment: $\Delta(3,11,\infty)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - 2 x^{9} - 2 x^{8} - 2 x^{7} - 2 x^{6} - 2 x^{5} - 2 x^{4} - 2 x^{3} - 2 x^{2} - 2 x - 1$.

[(3, 12, inf)]:

1.838647251857644507475889827820459296572281522911942692322792006225005665738703668210679250834536938

comment: $\Delta(3,12,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{10} - x^{9} - x^{7} - x^{6} - x^{4} - x^{3} - x - 1$.

[(3, inf, inf)]:

1.839286755214161132551852564653286600424178746097592246778758639404203222081966425738435419428307014

comment: $\Delta(3,\infty,\infty)$. The minimal polynomial of $\tau$ is $x^{3} - x^{2} - x - 1$. This is the tribonacci constant.

[(4, 4, inf)]:

1.839286755214161132551852564653286600424178746097592246778758639404203222081966425738435419428307014

comment: $\Delta(4,4,\infty)$. The minimal polynomial of $\tau$ is $x^{3} - x^{2} - x - 1$.

[(4, 5, inf)]:

1.885066078090561812262770852715130971458717572260482918179824253297444704764025578137301425141130200

comment: $\Delta(4,5,\infty)$. The minimal polynomial of $\tau$ is $x^{7} - x^{5} - 2 x^{4} - 3 x^{3} - 3 x^{2} - 2 x - 1$.

[(4, 6, inf)]:

1.906484761875724000834390202737388735766629277450784068378311381670259594725264769756103474790589353

comment: $\Delta(4,6,\infty)$. The minimal polynomial of $\tau$ is $x^{7} - x^{6} - 2 x^{4} - x^{3} - 2 x^{2} - x - 1$.

[(4, 7, inf)]:

1.916918159479150060997980959367486954946031759215200499911185933191134738870573966884342403934225780

comment: $\Delta(4,7,\infty)$. The minimal polynomial of $\tau$ is $x^{9} - x^{7} - 2 x^{6} - 3 x^{5} - 3 x^{4} - 3 x^{3} - 3 x^{2} - 2 x - 1$.

[(4, 8, inf)]:

1.922128004355224773675004886941362455038741003577474744311182062487992425380237963598384753506311527

comment: $\Delta(4,8,\infty)$. The minimal polynomial of $\tau$ is $x^{7} - x^{6} - x^{5} - x^{4} - x^{2} - x - 1$.

[(4, 9, inf)]:

1.924769469600623080900673170156996028635230269046094491599901706020537958749317300307839822104841892

comment: $\Delta(4,9,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{9} - 2 x^{8} - 3 x^{7} - 3 x^{6} - 3 x^{5} - 3 x^{4} - 3 x^{3} - 3 x^{2} - 2 x - 1$.

[(4, 10, inf)]:

1.926121254090910086054700253225233357371903490211620331833911021969040870352109772710998924376781378

comment: $\Delta(4,10,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{10} - 2 x^{8} - x^{7} - 2 x^{6} - x^{5} - 2 x^{4} - x^{3} - 2 x^{2} - x - 1$.

[(4, 11, inf)]:

1.926816937800952786910542412790010015880338110104773965320503396646061582361610938426393424882979213

comment: $\Delta(4,11,\infty)$. The minimal polynomial of $\tau$ is $x^{13} - x^{11} - 2 x^{10} - 3 x^{9} - 3 x^{8} - 3 x^{7} - 3 x^{6} - 3 x^{5} - 3 x^{4} - 3 x^{3} - 3 x^{2} - 2 x - 1$.

[(4, 12, inf)]:

1.927176168696939441119665948493757148736950099422691749492162315144086630857552029650520227669276163

comment: $\Delta(4,12,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{10} - x^{9} - x^{8} - x^{6} - x^{5} - x^{4} - x^{2} - x - 1$.

[(4, inf, inf)]:

1.927561975482925304261905861736622168698554255163384727146647038009666062297815559149818253461890653

comment: $\Delta(4,\infty,\infty)$. The minimal polynomial of $\tau$ is $x^{4} - x^{3} - x^{2} - x - 1$. This is the $4$-bonacci constant.

[(5, 5, inf)]:

1.927561975482925304261905861736622168698554255163384727146647038009666062297815559149818253461890653

comment: $\Delta(5,5,\infty)$. The minimal polynomial of $\tau$ is $x^{4} - x^{3} - x^{2} - x - 1$.

[(5, 6, inf)]:

1.947208609116713471697889066071509167138601656509465507978638391459024887569855343493454525789013837

comment: $\Delta(5,6,\infty)$. The minimal polynomial of $\tau$ is $x^{9} - x^{7} - 2 x^{6} - 3 x^{5} - 4 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(5, 7, inf)]:

1.956644457715633765898049511600728486591812463365434635985384931531438050416287466524609084209944750

comment: $\Delta(5,7,\infty)$. The minimal polynomial of $\tau$ is $x^{10} - x^{8} - 2 x^{7} - 3 x^{6} - 4 x^{5} - 4 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(5, 8, inf)]:

1.961282452241731427325139637236796333624773705360371785281131666188761974885674767980453619659335644

comment: $\Delta(5,8,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{9} - 2 x^{8} - 3 x^{7} - 4 x^{6} - 4 x^{5} - 4 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(5, 9, inf)]:

1.963594373572607908627408136322566459248267663732410564521910490892155004979832391387897299380128990

comment: $\Delta(5,9,\infty)$. The minimal polynomial of $\tau$ is $x^{12} - x^{10} - 2 x^{9} - 3 x^{8} - 4 x^{7} - 4 x^{6} - 4 x^{5} - 4 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(5, 10, inf)]:

1.964756554528429119388497882869970429445078369558699404660845082268630738176420436103522836937587732

comment: $\Delta(5,10,\infty)$. The minimal polynomial of $\tau$ is $x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{3} - x^{2} - x - 1$.

[(5, 11, inf)]:

1.965343697249660870297942137916881070709776289600301103694024453441433533580272332071252683389253533

comment: $\Delta(5,11,\infty)$. The minimal polynomial of $\tau$ is $x^{14} - x^{12} - 2 x^{11} - 3 x^{10} - 4 x^{9} - 4 x^{8} - 4 x^{7} - 4 x^{6} - 4 x^{5} - 4 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(5, 12, inf)]:

1.965641194885744197186424519244343136319718130819092607567433940704640372197605438875283150568932617

comment: $\Delta(5,12,\infty)$. The minimal polynomial of $\tau$ is $x^{15} - x^{13} - 2 x^{12} - 3 x^{11} - 4 x^{10} - 4 x^{9} - 4 x^{8} - 4 x^{7} - 4 x^{6} - 4 x^{5} - 4 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(5, inf, inf)]:

1.965948236645485337189937375934401396151327177456861393236934508442252712871886881734818665554630472

comment: $\Delta(5,\infty,\infty)$. The minimal polynomial of $\tau$ is $x^{5} - x^{4} - x^{3} - x^{2} - x - 1$. This is the $5$-bonacci constant.

[(6, 6, inf)]:

1.965948236645485337189937375934401396151327177456861393236934508442252712871886881734818665554630472

comment: $\Delta(6,6,\infty)$. The minimal polynomial of $\tau$ is $x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

[(6, 7, inf)]:

1.974893967276005571440101406637931987124010921424825315716372358467991546955891653179214094177389388

comment: $\Delta(6,7,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{9} - 2 x^{8} - 3 x^{7} - 4 x^{6} - 5 x^{5} - 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(6, 8, inf)]:

1.979260365169826598821522637628463675696357171156319850419615591410733731459729386558570173724081295

comment: $\Delta(6,8,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{10} - 2 x^{8} - x^{7} - 3 x^{6} - 2 x^{5} - 3 x^{4} - 2 x^{3} - 2 x^{2} - x - 1$.

[(6, 9, inf)]:

1.981420324857817586477520902325528214876426613983230202566013874295348668737515905892557514977144269

comment: $\Delta(6,9,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{10} - x^{9} - 2 x^{7} - 2 x^{6} - x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - x - 1$.

[(6, 10, inf)]:

1.982497368643498730602825350911677735648365400212014402369808062402924757955185317024081207089573077

comment: $\Delta(6,10,\infty)$. The minimal polynomial of $\tau$ is $x^{13} - x^{12} - 2 x^{10} - x^{9} - 3 x^{8} - 2 x^{7} - 3 x^{6} - 2 x^{5} - 3 x^{4} - 2 x^{3} - 2 x^{2} - x - 1$.

[(6, 11, inf)]:

1.983036953180574280525971663110240713530811200225662898191815786239949905194000210064658098721673394

comment: $\Delta(6,11,\infty)$. The minimal polynomial of $\tau$ is $x^{15} - x^{13} - 2 x^{12} - 3 x^{11} - 4 x^{10} - 5 x^{9} - 5 x^{8} - 5 x^{7} - 5 x^{6} - 5 x^{5} - 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(6, 12, inf)]:

1.983308015193266060147442779899934325290392930066423941584547764260754094462863011978411589456208355

comment: $\Delta(6,12,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{4} - x^{3} - x^{2} - x - 1$.

[(6, inf, inf)]:

1.983582843424326330385629293391425752730080865568821753216359065656702278014172402986575070226899797

comment: $\Delta(6,\infty,\infty)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$. This is the $6$-bonacci constant.

[(7, 7, inf)]:

1.983582843424326330385629293391425752730080865568821753216359065656702278014172402986575070226899797

comment: $\Delta(7,7,\infty)$. The minimal polynomial of $\tau$ is $x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

[(7, 8, inf)]:

1.987810356962827686022374983676082530419640387745031164186100324152771912102964537103377266255747487

comment: $\Delta(7,8,\infty)$. The minimal polynomial of $\tau$ is $x^{13} - x^{11} - 2 x^{10} - 3 x^{9} - 4 x^{8} - 5 x^{7} - 6 x^{6} - 6 x^{5} - 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(7, 9, inf)]:

1.989894193590286881875470475678197488902178594258782273883446055676495554747170543069602274080591961

comment: $\Delta(7,9,\infty)$. The minimal polynomial of $\tau$ is $x^{14} - x^{12} - 2 x^{11} - 3 x^{10} - 4 x^{9} - 5 x^{8} - 6 x^{7} - 6 x^{6} - 6 x^{5} - 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(7, 10, inf)]:

1.990929344817246400443243487230296246065286210293738255701700110357514144790517434826518792554451764

comment: $\Delta(7,10,\infty)$. The minimal polynomial of $\tau$ is $x^{15} - x^{13} - 2 x^{12} - 3 x^{11} - 4 x^{10} - 5 x^{9} - 6 x^{8} - 6 x^{7} - 6 x^{6} - 6 x^{5} - 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(7, 11, inf)]:

1.991445896973669314135055858091337436257896975964544746082140936886861971450057388612275452600427401

comment: $\Delta(7,11,\infty)$. The minimal polynomial of $\tau$ is $x^{16} - x^{14} - 2 x^{13} - 3 x^{12} - 4 x^{11} - 5 x^{10} - 6 x^{9} - 6 x^{8} - 6 x^{7} - 6 x^{6} - 6 x^{5} - 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(7, 12, inf)]:

1.991704339229072518989063956663352510773840906765106557359042705391597440182304808138629961442631520

comment: $\Delta(7,12,\infty)$. The minimal polynomial of $\tau$ is $x^{17} - x^{15} - 2 x^{14} - 3 x^{13} - 4 x^{12} - 5 x^{11} - 6 x^{10} - 6 x^{9} - 6 x^{8} - 6 x^{7} - 6 x^{6} - 6 x^{5} - 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(7, inf, inf)]:

1.991964196605035021097741754584374963479318960053157995244782153400951980309622183563141577022719017

comment: $\Delta(7,\infty,\infty)$. The minimal polynomial of $\tau$ is $x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$. This is the $7$-bonacci constant.

[(8, 8, inf)]:

1.991964196605035021097741754584374963479318960053157995244782153400951980309622183563141577022719017

comment: $\Delta(8,8,\infty)$. The minimal polynomial of $\tau$ is $x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

[(8, 9, inf)]:

1.994008279505726490367858497275734540772741962412684098791354459962295404327624667604946327047977422

comment: $\Delta(8,9,\infty)$. The minimal polynomial of $\tau$ is $x^{15} - x^{13} - 2 x^{12} - 3 x^{11} - 4 x^{10} - 5 x^{9} - 6 x^{8} - 7 x^{7} - 7 x^{6} - 6 x^{5} - 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(8, 10, inf)]:

1.995021841299653946408593133780679895618900417444805016437414606132491021444056431998586021784443907

comment: $\Delta(8,10,\infty)$. The minimal polynomial of $\tau$ is $x^{15} - x^{14} - 2 x^{12} - x^{11} - 3 x^{10} - 2 x^{9} - 4 x^{8} - 3 x^{7} - 4 x^{6} - 3 x^{5} - 3 x^{4} - 2 x^{3} - 2 x^{2} - x - 1$.

[(8, 11, inf)]:

1.995526659452187610514618937475503361647055382767991839180530806049425405074001049945330302614245907

comment: $\Delta(8,11,\infty)$. The minimal polynomial of $\tau$ is $x^{17} - x^{15} - 2 x^{14} - 3 x^{13} - 4 x^{12} - 5 x^{11} - 6 x^{10} - 7 x^{9} - 7 x^{8} - 7 x^{7} - 7 x^{6} - 6 x^{5} - 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(8, 12, inf)]:

1.995778737576971543204183669010944055495817187403226063436433279675760840912483226274734175468009612

comment: $\Delta(8,12,\infty)$. The minimal polynomial of $\tau$ is $x^{15} - x^{14} - x^{13} - x^{12} - 2 x^{10} - 2 x^{9} - 2 x^{8} - x^{7} - 2 x^{6} - 2 x^{5} - 2 x^{4} - x^{3} - x^{2} - x - 1$.

[(8, inf, inf)]:

1.996031179735414589815317901755839774489379680945221927634853627284595269945627520379784497496677719

comment: $\Delta(8,\infty,\infty)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$. This is the $8$-bonacci constant.

[(9, 9, inf)]:

1.996031179735414589815317901755839774489379680945221927634853627284595269945627520379784497496677719

comment: $\Delta(9,9,\infty)$. The minimal polynomial of $\tau$ is $x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

[(9, 10, inf)]:

1.997033357615845132541803805214927700542122013105266604447141755848408920165292322673203491873678343

comment: $\Delta(9,10,\infty)$. The minimal polynomial of $\tau$ is $x^{17} - x^{15} - 2 x^{14} - 3 x^{13} - 4 x^{12} - 5 x^{11} - 6 x^{10} - 7 x^{9} - 8 x^{8} - 8 x^{7} - 7 x^{6} - 6 x^{5} - 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(9, 11, inf)]:

1.997532043766426717280590594414384888659429643295114539976258895784694960712010059116290266928761504

comment: $\Delta(9,11,\infty)$. The minimal polynomial of $\tau$ is $x^{18} - x^{16} - 2 x^{15} - 3 x^{14} - 4 x^{13} - 5 x^{12} - 6 x^{11} - 7 x^{10} - 8 x^{9} - 8 x^{8} - 8 x^{7} - 7 x^{6} - 6 x^{5} - 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(9, 12, inf)]:

1.997780821999508156193845373418966315245417556263336953000729855858013167451933517656900964816344335

comment: $\Delta(9,12,\infty)$. The minimal polynomial of $\tau$ is $x^{17} - x^{16} - x^{15} - 2 x^{13} - 2 x^{12} - x^{11} - 3 x^{10} - 3 x^{9} - 2 x^{8} - 3 x^{7} - 3 x^{6} - 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - x - 1$.

[(9, inf, inf)]:

1.998029470262286698662331466971418894223948112007208176350076064183472462164183231134935063026401670

comment: $\Delta(9,\infty,\infty)$. The minimal polynomial of $\tau$ is $x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$. This is the $9$-bonacci constant.

[(10, 10, inf)]:

1.998029470262286698662331466971418894223948112007208176350076064183472462164183231134935063026401670

comment: $\Delta(10,10,\infty)$. The minimal polynomial of $\tau$ is $x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

[(10, 11, inf)]:

1.998524913224529490500993483767100239927894450356185894864123174447001192702370690690208675141273056

comment: $\Delta(10,11,\infty)$. The minimal polynomial of $\tau$ is $x^{19} - x^{17} - 2 x^{16} - 3 x^{15} - 4 x^{14} - 5 x^{13} - 6 x^{12} - 7 x^{11} - 8 x^{10} - 9 x^{9} - 9 x^{8} - 8 x^{7} - 7 x^{6} - 6 x^{5} - 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(10, 12, inf)]:

1.998771957208636801178021329246846951447114194873091587381054776840869616820234164240185544663645757

comment: $\Delta(10,12,\infty)$. The minimal polynomial of $\tau$ is $x^{19} - x^{18} - 2 x^{16} - x^{15} - 3 x^{14} - 2 x^{13} - 4 x^{12} - 3 x^{11} - 5 x^{10} - 4 x^{9} - 5 x^{8} - 4 x^{7} - 4 x^{6} - 3 x^{5} - 3 x^{4} - 2 x^{3} - 2 x^{2} - x - 1$.

[(10, inf, inf)]:

1.999018632710101138663409239129152861854310076062220826532008328824786927236540933057020104301232375

comment: $\Delta(10,\infty,\infty)$. The minimal polynomial of $\tau$ is $x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$. This is the $10$-bonacci constant.

[(11, 11, inf)]:

1.999018632710101138663409239129152861854310076062220826532008328824786927236540933057020104301232375

comment: $\Delta(11,11,\infty)$. The minimal polynomial of $\tau$ is $x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

[(11, 12, inf)]:

1.999264760006661860801578703588885433406321743983518687517524725782672592006793198003212774181072537

comment: $\Delta(11,12,\infty)$. The minimal polynomial of $\tau$ is $x^{21} - x^{19} - 2 x^{18} - 3 x^{17} - 4 x^{16} - 5 x^{15} - 6 x^{14} - 7 x^{13} - 8 x^{12} - 9 x^{11} - 10 x^{10} - 10 x^{9} - 9 x^{8} - 8 x^{7} - 7 x^{6} - 6 x^{5} - 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1$.

[(11, inf, inf)]:

1.999510401978285491440043956861529326661081254266614810779837519236615390078719789044140796520274345

comment: $\Delta(11,\infty,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$. This is the $11$-bonacci constant.

[(12, 12, inf)]:

1.999510401978285491440043956861529326661081254266614810779837519236615390078719789044140796520274345

comment: $\Delta(12,12,\infty)$. The minimal polynomial of $\tau$ is $x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

[(12, inf, inf)]:

1.999755500937317536697426762400382912383323386504556323062753939321611287661284370644702025866990915

comment: $\Delta(12,\infty,\infty)$. The minimal polynomial of $\tau$ is $x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$. This is the $12$-bonacci constant.