History of Characteristic polynomials of the classical test matrices

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2026-09-18 13:47 zeta3 repair T330 critique: tags, sources and formulas current reviewed
2026-09-18 13:25 zeta3 add measured range and exact verification notes
2026-09-18 13:25 zeta3 characteristic polynomials of classical test matrices for orders 2 through 12
2026-09-18 13:18 zeta3 drafted characteristic-polynomial test-matrix table prose

What changed between 2026-09-18 13:18 and 2026-09-18 13:25

from line 147 (1126 lines, 1124 more than before) @@ -147,2 +147,1126 @@
 Display properties:   number-header: $\chi_{A_n}(x)$+Numbers:+- params:+    family: hilbert+    n: '2'+  number: x^2 - 4/3*x + 1/12+- params:+    family: hilbert+    n: '3'+  number: x^3 - 23/15*x^2 + 127/720*x - 1/2160+- params:+    family: hilbert+    n: '4'+  number: x^4 - 176/105*x^3 + 3341/12600*x^2 - 41/23625*x + 1/6048000+- params:+    family: hilbert+    n: '5'+  number: x^5 - 563/315*x^4 + 735781/2116800*x^3 - 852401/222264000*x^2 + 61501/53343360000*x+    - 1/266716800000+- params:+    family: hilbert+    n: '6'+  number: x^6 - 6508/3465*x^5 + 14806217/34927200*x^4 - 18344719/2750517000*x^3 ++    10828423/2688505344000*x^2 - 3529/70573265280000*x + 1/186313420339200000+- params:+    family: hilbert+    n: '7'+  number: x^7 - 88069/45045*x^6 + 197708437/399567168*x^5 - 2804123885/276900047424*x^4+    + 126867217979/12687056718336000*x^3 - 509709971/1758426061161369600*x^2 + 8237351/59083115655022018560000*x+    - 1/2067909047925770649600000+- params:+    family: hilbert+    n: '8'+  number: x^8 - 91072/45045*x^7 + 6070382321/10821610800*x^6 - 85934726089/6093243231075*x^5+    + 3571285252517/176488178798131200*x^4 - 725120398661/668862621366980040000*x^3+    + 1258152740869/906142974022158943150080000*x^2 - 442037/17839689801061254193267200000*x+    + 1/365356847125734485878112256000000+- params:+    family: hilbert+    n: '9'+  number: x^9 - 1593269/765765*x^8 + 157170361853/252298126080*x^7 - 61871995671137/3342579143904000*x^6+    + 2734012728861723641/76447619528198510592000*x^5 - 53093363781067191073/17217915108238509548083200000*x^4+    + 93357540815902741633/11454734563208915632148791296000000*x^3 - 15631332539773/35919145033048771642891911168000000*x^2+    + 6384609533/22861817430634882175267843620208640000000*x - 1/1028781784378569697887052962909388800000000+- params:+    family: hilbert+    n: '10'+  number: x^10 - 31037876/14549535*x^9 + 971663633597021/1426115157667200*x^8 - 6354247328074737397/273017268427755852000*x^7+    + 160756576995060500777347/2798492525528920144404480000*x^6 - 943541357969761434451/129689398984002271506892800000*x^5+    + 55606538843763674112029693/1635346634651084049139354338164736000000*x^4 - 76258837068004289528537/18416047289464520248370554040657633280000000*x^3+    + 41204476395593715913/4667363025041888011747033216064270578483200000000*x^2 -+    114905419501/577586174348933641453695360487953484087296000000000*x + 1/46206893947914691316295628839036278726983680000000000+- params:+    family: hilbert+    n: '11'+  number: x^11 - 31730711/14549535*x^10 + 13304663823714511/18064125330451200*x^9+    - 7446978232543355472797/262824623739786300192000*x^8 + 5534509746813154007676733/64376983616217761905827840000*x^7+    - 70991538174569279610077271401/4734545428086162386576193216405504000*x^6 + 16938764763616402632019419305069/151656959152455953566808621107901104128000000*x^5+    - 537711699990639298115358228079/20494163175067135285250683013416215706337280000000*x^4+    + 24775226657018458411181475143/177246639402412628913430467137152019806456853299200000000*x^3+    - 13123781541297426135221/1140582124554525267057925056027573247454549850980352000000000*x^2+    + 26908963456783/3011136808823946705032922147912793373280011606588129280000000000*x+    - 1/33122504897063413755362143627040727106080127672469422080000000000+- params:+    family: hilbert+    n: '12'+  number: x^12 - 744355888/334639305*x^11 + 245819870293858261/311606161950283200*x^10+    - 2383120887400085792633/70839449367364276223625*x^9 + 2771974299490678101663092273/22743001063477937996223934464000*x^8+    - 25973541175774733587470747019519/930656503063432914198440631529708800000*x^7+    + 14812431968322121044485452691419016759/48164695742988698179294358275503034684735488000000*x^6+    - 143260817972982394003280010130132787/1152588694863429389414604508276268167354889717760000000*x^5+    + 2355395985981583363345207941591713093/1701260747083716258895654422947577842843681008402405785600000000*x^4+    - 1483525401061477340239167651383/4789580647024124930122172092604602745680850688967898038272000000000*x^3+    + 8822573393936250836151917/9205190837128606185700404988218638108978940556141182597715722240000000000*x^2+    - 12474672558791/493628358641021506708184217493224468593995687323070916802505605120000000000*x+    + 1/379106579436304517151885479034796391880188687864118464104324304732160000000000+- params:+    family: lehmer+    n: '2'+  number: x^2 - 2*x + 3/4+- params:+    family: lehmer+    n: '3'+  number: x^3 - 3*x^2 + 79/36*x - 5/12+- params:+    family: lehmer+    n: '4'+  number: x^4 - 4*x^3 + 311/72*x^2 - 29/18*x + 35/192+- params:+    family: lehmer+    n: '5'+  number: x^5 - 5*x^4 + 2563/360*x^3 - 283/72*x^2 + 4219/4800*x - 21/320+- params:+    family: lehmer+    n: '6'+  number: x^6 - 6*x^5 + 1271/120*x^4 - 463/60*x^3 + 332263/129600*x^2 - 8203/21600*x+    + 77/3840+- params:+    family: lehmer+    n: '7'+  number: x^7 - 7*x^6 + 12377/840*x^5 - 1597/120*x^4 + 37142527/6350400*x^3 - 167281/129600*x^2+    + 1147637/8467200*x - 143/26880+- params:+    family: lehmer+    n: '8'+  number: x^8 - 8*x^7 + 32839/1680*x^6 - 1473/70*x^5 + 291690043/25401600*x^4 - 2666407/793800*x^3+    + 5943727/11289600*x^2 - 11617/282240*x + 143/114688+- params:+    family: lehmer+    n: '9'+  number: x^9 - 9*x^8 + 378431/15120*x^7 - 52511/1680*x^6 + 4652605267/228614400*x^5+    - 20895043/2822400*x^4 + 12677280151/8230118400*x^3 - 54859373/304819200*x^2 ++    31810117/2926264320*x - 2431/9289728+- params:+    family: lehmer+    n: '10'+  number: x^10 - 10*x^9 + 471419/15120*x^8 - 66959/1512*x^7 + 7652477527/228614400*x^6+    - 332539783/22861440*x^5 + 4845758647/1285956000*x^4 - 2407635947/4115059200*x^3+    + 19335550709/365783040000*x^2 - 92874461/36578304000*x + 46189/928972800+- params:+    family: lehmer+    n: '11'+  number: x^11 - 11*x^10 + 6319609/166320*x^9 - 914209/15120*x^8 + 1438545223567/27662342400*x^7+    - 6003017357/228614400*x^6 + 10125142218371/1244805408000*x^5 - 4626106091/2939328000*x^4+    + 25109352035767/132779243520000*x^3 - 4972950919/365783040000*x^2 + 896112223/1686085632000*x+    - 4199/486604800+- params:+    family: lehmer+    n: '12'+  number: x^12 - 12*x^11 + 7564699/166320*x^10 - 1110559/13860*x^9 + 2136413366077/27662342400*x^8+    - 102364665547/2305195200*x^7 + 159298697098133/9958443264000*x^6 - 1024581940741/276623424000*x^5+    + 994796760264497/1792519787520000*x^4 - 15860857202669/298753297920000*x^3 ++    9903514190701/3186701844480000*x^2 - 764568997/7587385344000*x + 96577/70071091200+- params:+    family: pascal+    n: '2'+  number: x^2 - 3*x + 1+- params:+    family: pascal+    n: '3'+  number: x^3 - 9*x^2 + 9*x - 1+- params:+    family: pascal+    n: '4'+  number: x^4 - 29*x^3 + 72*x^2 - 29*x + 1+- params:+    family: pascal+    n: '5'+  number: x^5 - 99*x^4 + 626*x^3 - 626*x^2 + 99*x - 1+- params:+    family: pascal+    n: '6'+  number: x^6 - 351*x^5 + 6084*x^4 - 13869*x^3 + 6084*x^2 - 351*x + 1+- params:+    family: pascal+    n: '7'+  number: x^7 - 1275*x^6 + 64974*x^5 - 347020*x^4 + 347020*x^3 - 64974*x^2 + 1275*x+    - 1+- params:+    family: pascal+    n: '8'+  number: x^8 - 4707*x^7 + 744193*x^6 - 9952274*x^5 + 21537270*x^4 - 9952274*x^3 ++    744193*x^2 - 4707*x + 1+- params:+    family: pascal+    n: '9'+  number: x^9 - 17577*x^8 + 8965323*x^7 - 321541977*x^6 + 1545936516*x^5 - 1545936516*x^4+    + 321541977*x^3 - 8965323*x^2 + 17577*x - 1+- params:+    family: pascal+    n: '10'+  number: x^10 - 66197*x^9 + 112088583*x^8 - 11416400590*x^7 + 129464229047*x^6 -+    270190791369*x^5 + 129464229047*x^4 - 11416400590*x^3 + 112088583*x^2 - 66197*x+    + 1+- params:+    family: pascal+    n: '11'+  number: x^11 - 250953*x^10 + 1441465015*x^9 - 435869304863*x^8 + 12439806093822*x^7+    - 55538046772046*x^6 + 55538046772046*x^5 - 12439806093822*x^4 + 435869304863*x^3+    - 1441465015*x^2 + 250953*x - 1+- params:+    family: pascal+    n: '12'+  number: x^12 - 956385*x^11 + 18952951005*x^10 - 17605464402686*x^9 + 1338642053600985*x^8+    - 13499435968309125*x^7 + 27414197906689626*x^6 - 13499435968309125*x^5 + 1338642053600985*x^4+    - 17605464402686*x^3 + 18952951005*x^2 - 956385*x + 1+- params:+    family: redheffer+    n: '2'+  number: x^2 - 2*x+- params:+    family: redheffer+    n: '3'+  number: x^3 - 3*x^2 + x + 1+- params:+    family: redheffer+    n: '4'+  number: x^4 - 4*x^3 + 3*x^2 + x - 1+- params:+    family: redheffer+    n: '5'+  number: x^5 - 5*x^4 + 6*x^3 + x^2 - 5*x + 2+- params:+    family: redheffer+    n: '6'+  number: x^6 - 6*x^5 + 10*x^4 - 3*x^3 - 6*x^2 + 5*x - 1+- params:+    family: redheffer+    n: '7'+  number: x^7 - 7*x^6 + 15*x^5 - 8*x^4 - 13*x^3 + 21*x^2 - 11*x + 2+- params:+    family: redheffer+    n: '8'+  number: x^8 - 8*x^7 + 21*x^6 - 19*x^5 - 11*x^4 + 38*x^3 - 33*x^2 + 13*x - 2+- params:+    family: redheffer+    n: '9'+  number: x^9 - 9*x^8 + 28*x^7 - 34*x^6 - 7*x^5 + 69*x^4 - 86*x^3 + 52*x^2 - 16*x+    + 2+- params:+    family: redheffer+    n: '10'+  number: x^10 - 10*x^9 + 36*x^8 - 56*x^7 + 13*x^6 + 90*x^5 - 155*x^4 + 124*x^3 -+    54*x^2 + 12*x - 1+- params:+    family: redheffer+    n: '11'+  number: x^11 - 11*x^10 + 45*x^9 - 83*x^8 + 33*x^7 + 161*x^6 - 371*x^5 + 405*x^4+    - 262*x^3 + 102*x^2 - 22*x + 2+- params:+    family: redheffer+    n: '12'+  number: x^12 - 12*x^11 + 55*x^10 - 122*x^9 + 104*x^8 + 128*x^7 - 490*x^6 + 692*x^5+    - 583*x^4 + 316*x^3 - 109*x^2 + 22*x - 2+- params:+    family: vandermonde+    n: '2'+  number: x^2 - 3*x + 1+- params:+    family: vandermonde+    n: '3'+  number: x^3 - 12*x^2 + 15*x - 2+- params:+    family: vandermonde+    n: '4'+  number: x^4 - 76*x^3 + 318*x^2 - 206*x + 12+- params:+    family: vandermonde+    n: '5'+  number: x^5 - 701*x^4 + 13712*x^3 - 29654*x^2 + 10644*x - 288+- params:+    family: vandermonde+    n: '6'+  number: x^6 - 8477*x^5 + 1184563*x^4 - 9879696*x^3 + 12780924*x^2 - 2621088*x ++    34560+- params:+    family: vandermonde+    n: '7'+  number: x^7 - 126126*x^6 + 175391021*x^5 - 8383939410*x^4 + 37789054920*x^3 - 31167845568*x^2+    + 3722872320*x - 24883200+- params:+    family: vandermonde+    n: '8'+  number: x^8 - 2223278*x^7 + 39292455532*x^6 - 15649805095100*x^5 + 336801882794736*x^4+    - 935611268352768*x^3 + 506900977274880*x^2 - 35802346291200*x + 125411328000+- params:+    family: vandermonde+    n: '9'+  number: x^9 - 45269999*x^8 + 12333773739126*x^7 - 54119093224422012*x^6 + 8233381582549148304*x^5+    - 94721550692814747648*x^4 + 175077442934730961920*x^3 - 63238986969641164800*x^2+    + 2676769226514432000*x - 5056584744960000+- params:+    family: vandermonde+    n: '10'+  number: x^10 - 1045269999*x^9 + 5156725421771705*x^8 - 308907278210229422688*x^7+    + 453497408002063611276912*x^6 - 31684712236678200354462720*x^5 + 219012031673412626436203520*x^4+    - 281757108425049655288012800*x^3 + 68398369246543120146432000*x^2 - 1756842252383252643840000*x+    + 1834933472251084800000+- params:+    family: vandermonde+    n: '11'+  number: x^11 - 26982694600*x^10 + 2768433904496830779*x^9 - 2698462454434569862116586*x^8+    + 48357738246807058330220466984*x^7 - 28774167853086758094066820442880*x^6 + 1052216274928656962820168339033600*x^5+    - 4744908475150754942696710814515200*x^4 + 4361829180433097016225586765824000*x^3+    - 715224379087980740071076659200000*x^2 + 11280390867268875273948364800000*x -+    6658606584104736522240000000+- params:+    family: vandermonde+    n: '12'+  number: x^12 - 769991065288*x^11 + 1855604653961893375754*x^10 - 34176022967059952707187048538*x^9+    + 9030338952054561183393619390435068*x^8 - 59082763135905281911358141137088260128*x^7+    + 16334313261503004964235814052132092902400*x^6 - 344979275304455476260326285722400024064000*x^5+    + 1076090262741617342861267885208604483584000*x^4 - 717535900366316538606114377052192768000000*x^3+    + 79777760171097248230224390699928780800000*x^2 - 781325878741395601622544899112960000000*x+    + 265790267296391946810949632000000000+- params:+    family: cauchy+    n: '2'+  number: x^2 - 3/4*x + 1/72+- params:+    family: cauchy+    n: '3'+  number: x^3 - 11/12*x^2 + 131/3600*x - 1/43200+- params:+    family: cauchy+    n: '4'+  number: x^4 - 25/24*x^3 + 22151/352800*x^2 - 19/169344*x + 1/423360000+- params:+    family: cauchy+    n: '5'+  number: x^5 - 137/120*x^4 + 576593/6350400*x^3 - 227491/762048000*x^2 + 48917/2240421120000*x+    - 1/67212633600000+- params:+    family: cauchy+    n: '6'+  number: x^6 - 49/40*x^5 + 45831857/384199200*x^4 - 187687/313632000*x^3 + 92510189/975927439872000*x^2+    - 71/265558487040000*x + 1/172153600393420800000+- params:+    family: cauchy+    n: '7'+  number: x^7 - 363/280*x^6 + 1278639679/8657288640*x^5 - 672728317/661102041600*x^4+    + 1863308364413/6652246739314176000*x^3 - 200163947/101597950200434688000*x^2+    + 5212451/25346656616004445962240000*x - 1/7097063852481244869427200000+- params:+    family: cauchy+    n: '8'+  number: x^8 - 761/560*x^7 + 45633534571/259718659200*x^6 - 37700568931/24240408192000*x^5+    + 1050802989214003/1606042427062993920000*x^4 - 4040992946279/449691879577638297600000*x^3+    + 13725322878221/5183137811406749154818457600000*x^2 - 14402903/1451278587193889763349168128000000*x+    + 1/4702142622508202833251304734720000000+- params:+    family: cauchy+    n: '9'+  number: x^9 - 7129/5040*x^8 + 4356851329043/21445340716800*x^7 - 238568988351797/108084517212672000*x^6+    + 16960514678235618433/12996095319793746800640000*x^5 - 494968591335137683/16375080102940120968806400000*x^4+    + 24458864498578595287/1265748169234585177352441438208000000*x^3 - 2913737409515171/12758741545884618587712609697136640000000*x^2+    + 16687129889/555770781738733985680761278407272038400000000*x - 1/50019370356486058711268515056654483456000000000+- params:+    family: cauchy+    n: '10'+  number: x^10 - 7381/5040*x^9 + 1557959244945911/6774046998919200*x^8 - 3349130962422473/1128634607423232000*x^7+    + 2102006513410124714624941/903913085745841206642647040000*x^6 - 3101489430454625873293/37650594645942476706437529600000*x^5+    + 205458780408479711146497241/2112867851969200591488045804908838912000000*x^4+    - 120728732319564349049/44003528817870954467354342383225405440000000*x^3 + 49264137944540225359/39196514684301775522651584948507744318101913600000000*x^2+    - 331868947/58308864819622475984109795791168545266597888000000000*x + 1/8537000898240926708833515201784986712482596782080000000000+- params:+    family: cauchy+    n: '11'+  number: x^11 - 83711/55440*x^10 + 1542425044049287/6021375110150400*x^9 - 1278197836195064479/333825036106738176000*x^8+    + 11187274328720652555546796777/2936813615588238080381960232960000*x^7 - 6268711589330820110859317353/32563389369642383835275175063060480000*x^6+    + 18428715687034746818545456295917/48985197806243273002079184617852056633344000000*x^5+    - 19295243516856341769104357399/905246455459375685078423331737906006584197120000000*x^4+    + 2913776331130613461980920573/114501329053958558278076081770600204794971127231283200000000*x^3+    - 697802097165541896349/1586988420687865617734134493340518838458299823425585152000000000*x^2+    + 3986898585601/59004229481174843667355120462400490413879587434963255951360000000000*x+    - 1/23365674874545238092272627703110594203896316624245449356738560000000000+- params:+    family: cauchy+    n: '12'+  number: x^12 - 86021/55440*x^11 + 20598502049778241/73132058416903200*x^10 - 1607068119580002727/335870878466648448000*x^9+    + 1019682547918716415880594009/174363008153330857971050164224000*x^8 - 3091201111273342205298885563/7721908273507630020109632243302400000*x^7+    + 25193028406046305766309936156170327603/21047972039686061104351634566394826157229408256000000*x^6+    - 834906667109738906928594132542753/6904731182723048684173104262490705101519517122560000000*x^5+    + 29777379593844416050873110263293949/95027564587104722461171554196071848078839896326477237452800000000*x^4+    - 9061442784202024339420427227/592297251085636866578104546319374212380631912393242975404032000000000*x^3+    + 313771206516880435280147/32181347166601607225208615838812358828990376184269574361614164951040000000000*x^2+    - 150225179491/2991151486940698498778561760923458772105212006523706947172989958881280000000000*x+    + 1/1025163331422159477883374029444818871881163521419883129418493194587298856960000000000+- params:+    family: fiedler+    n: '2'+  number: x^2 - 1+- params:+    family: fiedler+    n: '3'+  number: x^3 - 6*x - 4+- params:+    family: fiedler+    n: '4'+  number: x^4 - 20*x^2 - 32*x - 12+- params:+    family: fiedler+    n: '5'+  number: x^5 - 50*x^3 - 140*x^2 - 120*x - 32+- params:+    family: fiedler+    n: '6'+  number: x^6 - 105*x^4 - 448*x^3 - 648*x^2 - 384*x - 80+- params:+    family: fiedler+    n: '7'+  number: x^7 - 196*x^5 - 1176*x^4 - 2520*x^3 - 2464*x^2 - 1120*x - 192+- params:+    family: fiedler+    n: '8'+  number: x^8 - 336*x^6 - 2688*x^5 - 7920*x^4 - 11264*x^3 - 8320*x^2 - 3072*x - 448+- params:+    family: fiedler+    n: '9'+  number: x^9 - 540*x^7 - 5544*x^6 - 21384*x^5 - 41184*x^4 - 43680*x^3 - 25920*x^2+    - 8064*x - 1024+- params:+    family: fiedler+    n: '10'+  number: x^10 - 825*x^8 - 10560*x^7 - 51480*x^6 - 128128*x^5 - 182000*x^4 - 153600*x^3+    - 76160*x^2 - 20480*x - 2304+- params:+    family: fiedler+    n: '11'+  number: x^11 - 1210*x^9 - 18876*x^8 - 113256*x^7 - 352352*x^6 - 640640*x^5 - 718080*x^4+    - 502656*x^3 - 214016*x^2 - 50688*x - 5120+- params:+    family: fiedler+    n: '12'+  number: x^12 - 1716*x^10 - 32032*x^9 - 231660*x^8 - 878592*x^7 - 1980160*x^6 - 2820096*x^5+    - 2604672*x^4 - 1556480*x^3 - 580608*x^2 - 122880*x - 11264+- params:+    family: kms+    n: '2'+  number: x^2 - 2*x + 3/4+- params:+    family: kms+    n: '3'+  number: x^3 - 3*x^2 + 39/16*x - 9/16+- params:+    family: kms+    n: '4'+  number: x^4 - 4*x^3 + 327/64*x^2 - 81/32*x + 27/64+- params:+    family: kms+    n: '5'+  number: x^5 - 5*x^4 + 2247/256*x^3 - 1755/256*x^2 + 621/256*x - 81/256+- params:+    family: kms+    n: '6'+  number: x^6 - 6*x^5 + 13767/1024*x^4 - 3717/256*x^3 + 4131/512*x^2 - 567/256*x ++    243/1024+- params:+    family: kms+    n: '7'+  number: x^7 - 7*x^6 + 78279/4096*x^5 - 108621/4096*x^4 + 41715/2048*x^3 - 17901/2048*x^2+    + 8019/4096*x - 729/4096+- params:+    family: kms+    n: '8'+  number: x^8 - 8*x^7 + 422343/16384*x^6 - 359235/8192*x^5 + 709965/16384*x^4 - 105057/4096*x^3+    + 146529/16384*x^2 - 13851/8192*x + 2187/16384+- params:+    family: kms+    n: '9'+  number: x^9 - 9*x^8 + 2191815/65536*x^7 - 4424895/65536*x^6 + 5371515/65536*x^5+    - 4087827/65536*x^4 + 1961253/65536*x^3 - 575181/65536*x^2 + 94041/65536*x - 6561/65536+- params:+    family: kms+    n: '10'+  number: x^10 - 10*x^9 + 11039175/262144*x^8 - 3228255/32768*x^7 + 2330775/16384*x^6+    - 544887/4096*x^5 + 10741815/131072*x^4 - 1081107/32768*x^3 + 273375/32768*x^2+    - 19683/16384*x + 19683/262144+- params:+    family: kms+    n: '11'+  number: x^11 - 11*x^10 + 54292935/1048576*x^9 - 144548145/1048576*x^8 + 60613515/262144*x^7+    - 67472433/262144*x^6 + 102095721/524288*x^5 - 52697223/524288*x^4 + 9126351/262144*x^3+    - 2027349/262144*x^2 + 1043199/1048576*x - 59049/1048576+- params:+    family: kms+    n: '12'+  number: x^12 - 12*x^11 + 261910983/4194304*x^10 - 391157685/2097152*x^9 + 1496745135/4194304*x^8+    - 242164971/524288*x^7 + 872345943/2097152*x^6 - 276237783/1048576*x^5 + 244981179/2097152*x^4+    - 18600435/524288*x^3 + 29465451/4194304*x^2 - 1712421/2097152*x + 177147/4194304+- params:+    family: minij+    n: '2'+  number: x^2 - 3*x + 1+- params:+    family: minij+    n: '3'+  number: x^3 - 6*x^2 + 5*x - 1+- params:+    family: minij+    n: '4'+  number: x^4 - 10*x^3 + 15*x^2 - 7*x + 1+- params:+    family: minij+    n: '5'+  number: x^5 - 15*x^4 + 35*x^3 - 28*x^2 + 9*x - 1+- params:+    family: minij+    n: '6'+  number: x^6 - 21*x^5 + 70*x^4 - 84*x^3 + 45*x^2 - 11*x + 1+- params:+    family: minij+    n: '7'+  number: x^7 - 28*x^6 + 126*x^5 - 210*x^4 + 165*x^3 - 66*x^2 + 13*x - 1+- params:+    family: minij+    n: '8'+  number: x^8 - 36*x^7 + 210*x^6 - 462*x^5 + 495*x^4 - 286*x^3 + 91*x^2 - 15*x + 1+- params:+    family: minij+    n: '9'+  number: x^9 - 45*x^8 + 330*x^7 - 924*x^6 + 1287*x^5 - 1001*x^4 + 455*x^3 - 120*x^2+    + 17*x - 1+- params:+    family: minij+    n: '10'+  number: x^10 - 55*x^9 + 495*x^8 - 1716*x^7 + 3003*x^6 - 3003*x^5 + 1820*x^4 - 680*x^3+    + 153*x^2 - 19*x + 1+- params:+    family: minij+    n: '11'+  number: x^11 - 66*x^10 + 715*x^9 - 3003*x^8 + 6435*x^7 - 8008*x^6 + 6188*x^5 - 3060*x^4+    + 969*x^3 - 190*x^2 + 21*x - 1+- params:+    family: minij+    n: '12'+  number: x^12 - 78*x^11 + 1001*x^10 - 5005*x^9 + 12870*x^8 - 19448*x^7 + 18564*x^6+    - 11628*x^5 + 4845*x^4 - 1330*x^3 + 231*x^2 - 23*x + 1+- params:+    family: grcar+    n: '2'+  number: x^2 - 2*x + 2+- params:+    family: grcar+    n: '3'+  number: x^3 - 3*x^2 + 5*x - 4+- params:+    family: grcar+    n: '4'+  number: x^4 - 4*x^3 + 9*x^2 - 12*x + 8+- params:+    family: grcar+    n: '5'+  number: x^5 - 5*x^4 + 14*x^3 - 25*x^2 + 28*x - 15+- params:+    family: grcar+    n: '6'+  number: x^6 - 6*x^5 + 20*x^4 - 44*x^3 + 66*x^2 - 62*x + 29+- params:+    family: grcar+    n: '7'+  number: x^7 - 7*x^6 + 27*x^5 - 70*x^4 + 129*x^3 - 165*x^2 + 136*x - 56+- params:+    family: grcar+    n: '8'+  number: x^8 - 8*x^7 + 35*x^6 - 104*x^5 + 225*x^4 - 356*x^3 + 401*x^2 - 294*x + 108+- params:+    family: grcar+    n: '9'+  number: x^9 - 9*x^8 + 44*x^7 - 147*x^6 + 363*x^5 - 676*x^4 + 944*x^3 - 951*x^2 ++    628*x - 208+- params:+    family: grcar+    n: '10'+  number: x^10 - 10*x^9 + 54*x^8 - 200*x^7 + 553*x^6 - 1176*x^5 + 1935*x^4 - 2424*x^3+    + 2211*x^2 - 1328*x + 401+- params:+    family: grcar+    n: '11'+  number: x^11 - 11*x^10 + 65*x^9 - 264*x^8 + 806*x^7 - 1918*x^6 + 3605*x^5 - 5330*x^4+    + 6064*x^3 - 5056*x^2 + 2787*x - 773+- params:+    family: grcar+    n: '12'+  number: x^12 - 12*x^11 + 77*x^10 - 340*x^9 + 1134*x^8 - 2976*x^7 + 6258*x^6 - 10578*x^5+    + 14230*x^4 - 14844*x^3 + 11406*x^2 - 5810*x + 1490+- params:+    family: chow+    n: '2'+  number: x^2 - 2*x+- params:+    family: chow+    n: '3'+  number: x^3 - 3*x^2 + x+- params:+    family: chow+    n: '4'+  number: x^4 - 4*x^3 + 3*x^2+- params:+    family: chow+    n: '5'+  number: x^5 - 5*x^4 + 6*x^3 - x^2+- params:+    family: chow+    n: '6'+  number: x^6 - 6*x^5 + 10*x^4 - 4*x^3+- params:+    family: chow+    n: '7'+  number: x^7 - 7*x^6 + 15*x^5 - 10*x^4 + x^3+- params:+    family: chow+    n: '8'+  number: x^8 - 8*x^7 + 21*x^6 - 20*x^5 + 5*x^4+- params:+    family: chow+    n: '9'+  number: x^9 - 9*x^8 + 28*x^7 - 35*x^6 + 15*x^5 - x^4+- params:+    family: chow+    n: '10'+  number: x^10 - 10*x^9 + 36*x^8 - 56*x^7 + 35*x^6 - 6*x^5+- params:+    family: chow+    n: '11'+  number: x^11 - 11*x^10 + 45*x^9 - 84*x^8 + 70*x^7 - 21*x^6 + x^5+- params:+    family: chow+    n: '12'+  number: x^12 - 12*x^11 + 55*x^10 - 120*x^9 + 126*x^8 - 56*x^7 + 7*x^6+- params:+    family: frank+    n: '2'+  number: x^2 - 3*x + 1+- params:+    family: frank+    n: '3'+  number: x^3 - 6*x^2 + 6*x - 1+- params:+    family: frank+    n: '4'+  number: x^4 - 10*x^3 + 21*x^2 - 10*x + 1+- params:+    family: frank+    n: '5'+  number: x^5 - 15*x^4 + 55*x^3 - 55*x^2 + 15*x - 1+- params:+    family: frank+    n: '6'+  number: x^6 - 21*x^5 + 120*x^4 - 215*x^3 + 120*x^2 - 21*x + 1+- params:+    family: frank+    n: '7'+  number: x^7 - 28*x^6 + 231*x^5 - 665*x^4 + 665*x^3 - 231*x^2 + 28*x - 1+- params:+    family: frank+    n: '8'+  number: x^8 - 36*x^7 + 406*x^6 - 1736*x^5 + 2835*x^4 - 1736*x^3 + 406*x^2 - 36*x+    + 1+- params:+    family: frank+    n: '9'+  number: x^9 - 45*x^8 + 666*x^7 - 3990*x^6 + 9891*x^5 - 9891*x^4 + 3990*x^3 - 666*x^2+    + 45*x - 1+- params:+    family: frank+    n: '10'+  number: x^10 - 55*x^9 + 1035*x^8 - 8310*x^7 + 29505*x^6 - 45297*x^5 + 29505*x^4+    - 8310*x^3 + 1035*x^2 - 55*x + 1+- params:+    family: frank+    n: '11'+  number: x^11 - 66*x^10 + 1540*x^9 - 16005*x^8 + 77715*x^7 - 173712*x^6 + 173712*x^5+    - 77715*x^4 + 16005*x^3 - 1540*x^2 + 66*x - 1+- params:+    family: frank+    n: '12'+  number: x^12 - 78*x^11 + 2211*x^10 - 28930*x^9 + 185130*x^8 - 575982*x^7 + 845691*x^6+    - 575982*x^5 + 185130*x^4 - 28930*x^3 + 2211*x^2 - 78*x + 1+- params:+    family: wilkinson+    n: '2'+  number: x^2 - x - 3/4+- params:+    family: wilkinson+    n: '3'+  number: x^3 - 2*x^2 - x + 2+- params:+    family: wilkinson+    n: '4'+  number: x^4 - 4*x^3 + 5/2*x^2 + 4*x - 35/16+- params:+    family: wilkinson+    n: '5'+  number: x^5 - 6*x^4 + 9*x^3 + 4*x^2 - 13*x + 4+- params:+    family: wilkinson+    n: '6'+  number: x^6 - 9*x^5 + 107/4*x^4 - 47/2*x^3 - 281/16*x^2 + 487/16*x - 403/64+- params:+    family: wilkinson+    n: '7'+  number: x^7 - 12*x^6 + 52*x^5 - 90*x^4 + 17*x^3 + 118*x^2 - 106*x + 20+- params:+    family: wilkinson+    n: '8'+  number: x^8 - 16*x^7 + 100*x^6 - 301*x^5 + 3257/8*x^4 - 117/2*x^3 - 3251/8*x^2 ++    5275/16*x - 13419/256+- params:+    family: wilkinson+    n: '9'+  number: x^9 - 20*x^8 + 162*x^7 - 672*x^6 + 1446*x^5 - 1218*x^4 - 867*x^3 + 2468*x^2+    - 1547*x + 252+- params:+    family: wilkinson+    n: '10'+  number: x^10 - 25*x^9 + 1049/4*x^8 - 1491*x^7 + 39321/8*x^6 - 72723/8*x^5 + 233253/32*x^4+    + 49189/16*x^3 - 2621331/256*x^2 + 1595895/256*x - 925699/1024+- params:+    family: wilkinson+    n: '11'+  number: x^11 - 30*x^10 + 385*x^9 - 2750*x^8 + 11859*x^7 - 30922*x^6 + 43923*x^5+    - 16030*x^4 - 46093*x^3 + 71218*x^2 - 37099*x + 5610+- params:+    family: wilkinson+    n: '12'+  number: x^12 - 36*x^11 + 1131/2*x^10 - 5080*x^9 + 458923/16*x^8 - 209835/2*x^7 ++    3912537/16*x^6 - 1320333/4*x^5 + 40361063/256*x^4 + 13153015/64*x^3 - 181261805/512*x^2+    + 11688285/64*x - 102416243/4096+- params:+    family: clement+    n: '2'+  number: x^2 - 1+- params:+    family: clement+    n: '3'+  number: x^3 - 4*x+- params:+    family: clement+    n: '4'+  number: x^4 - 10*x^2 + 9+- params:+    family: clement+    n: '5'+  number: x^5 - 20*x^3 + 64*x+- params:+    family: clement+    n: '6'+  number: x^6 - 35*x^4 + 259*x^2 - 225+- params:+    family: clement+    n: '7'+  number: x^7 - 56*x^5 + 784*x^3 - 2304*x+- params:+    family: clement+    n: '8'+  number: x^8 - 84*x^6 + 1974*x^4 - 12916*x^2 + 11025+- params:+    family: clement+    n: '9'+  number: x^9 - 120*x^7 + 4368*x^5 - 52480*x^3 + 147456*x+- params:+    family: clement+    n: '10'+  number: x^10 - 165*x^8 + 8778*x^6 - 172810*x^4 + 1057221*x^2 - 893025+- params:+    family: clement+    n: '11'+  number: x^11 - 220*x^9 + 16368*x^7 - 489280*x^5 + 5395456*x^3 - 14745600*x+- params:+    family: clement+    n: '12'+  number: x^12 - 286*x^10 + 28743*x^8 - 1234948*x^6 + 21967231*x^4 - 128816766*x^2+    + 108056025+- params:+    family: moler+    n: '2'+  number: x^2 - 3*x + 1+- params:+    family: moler+    n: '3'+  number: x^3 - 6*x^2 + 9*x - 1+- params:+    family: moler+    n: '4'+  number: x^4 - 10*x^3 + 31*x^2 - 31*x + 1+- params:+    family: moler+    n: '5'+  number: x^5 - 15*x^4 + 75*x^3 - 156*x^2 + 117*x - 1+- params:+    family: moler+    n: '6'+  number: x^6 - 21*x^5 + 150*x^4 - 492*x^3 + 765*x^2 - 459*x + 1+- params:+    family: moler+    n: '7'+  number: x^7 - 28*x^6 + 266*x^5 - 1218*x^4 + 2949*x^3 - 3650*x^2 + 1825*x - 1+- params:+    family: moler+    n: '8'+  number: x^8 - 36*x^7 + 434*x^6 - 2590*x^5 + 8655*x^4 - 16558*x^3 + 17003*x^2 - 7287*x+    + 1+- params:+    family: moler+    n: '9'+  number: x^9 - 45*x^8 + 666*x^7 - 4956*x^6 + 21375*x^5 - 56073*x^4 + 88575*x^3 -+    77688*x^2 + 29133*x - 1+- params:+    family: moler+    n: '10'+  number: x^10 - 55*x^9 + 975*x^8 - 8772*x^7 + 46779*x^6 - 157483*x^5 + 339660*x^4+    - 456720*x^3 + 349545*x^2 - 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