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Properties

Label 28.0.26333545371...9456.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{42}\cdot 3^{14}\cdot 29^{24}$
Root discriminant $87.82$
Ramified primes $2, 3, 29$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{14}$ (as 28T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![289, -5746, 155537, 898658, 5020235, 9907754, 21872382, 8191560, 37771153, 6404386, 37654712, 1057018, 25100954, -371480, 11555688, -842760, 3760649, -341770, 872712, -97228, 139789, -12210, 14548, -1134, 1043, -50, 41, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 2*x^27 + 41*x^26 - 50*x^25 + 1043*x^24 - 1134*x^23 + 14548*x^22 - 12210*x^21 + 139789*x^20 - 97228*x^19 + 872712*x^18 - 341770*x^17 + 3760649*x^16 - 842760*x^15 + 11555688*x^14 - 371480*x^13 + 25100954*x^12 + 1057018*x^11 + 37654712*x^10 + 6404386*x^9 + 37771153*x^8 + 8191560*x^7 + 21872382*x^6 + 9907754*x^5 + 5020235*x^4 + 898658*x^3 + 155537*x^2 - 5746*x + 289)
 
gp: K = bnfinit(x^28 - 2*x^27 + 41*x^26 - 50*x^25 + 1043*x^24 - 1134*x^23 + 14548*x^22 - 12210*x^21 + 139789*x^20 - 97228*x^19 + 872712*x^18 - 341770*x^17 + 3760649*x^16 - 842760*x^15 + 11555688*x^14 - 371480*x^13 + 25100954*x^12 + 1057018*x^11 + 37654712*x^10 + 6404386*x^9 + 37771153*x^8 + 8191560*x^7 + 21872382*x^6 + 9907754*x^5 + 5020235*x^4 + 898658*x^3 + 155537*x^2 - 5746*x + 289, 1)
 

Normalized defining polynomial

\( x^{28} - 2 x^{27} + 41 x^{26} - 50 x^{25} + 1043 x^{24} - 1134 x^{23} + 14548 x^{22} - 12210 x^{21} + 139789 x^{20} - 97228 x^{19} + 872712 x^{18} - 341770 x^{17} + 3760649 x^{16} - 842760 x^{15} + 11555688 x^{14} - 371480 x^{13} + 25100954 x^{12} + 1057018 x^{11} + 37654712 x^{10} + 6404386 x^{9} + 37771153 x^{8} + 8191560 x^{7} + 21872382 x^{6} + 9907754 x^{5} + 5020235 x^{4} + 898658 x^{3} + 155537 x^{2} - 5746 x + 289 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 14]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2633354537185343913419055876339673429183385802381459456=2^{42}\cdot 3^{14}\cdot 29^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.82$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(696=2^{3}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{696}(1,·)$, $\chi_{696}(277,·)$, $\chi_{696}(517,·)$, $\chi_{696}(257,·)$, $\chi_{696}(545,·)$, $\chi_{696}(397,·)$, $\chi_{696}(629,·)$, $\chi_{696}(401,·)$, $\chi_{696}(661,·)$, $\chi_{696}(25,·)$, $\chi_{696}(281,·)$, $\chi_{696}(413,·)$, $\chi_{696}(197,·)$, $\chi_{696}(161,·)$, $\chi_{696}(373,·)$, $\chi_{696}(65,·)$, $\chi_{696}(529,·)$, $\chi_{696}(169,·)$, $\chi_{696}(581,·)$, $\chi_{696}(605,·)$, $\chi_{696}(49,·)$, $\chi_{696}(625,·)$, $\chi_{696}(181,·)$, $\chi_{696}(349,·)$, $\chi_{696}(233,·)$, $\chi_{696}(313,·)$, $\chi_{696}(509,·)$, $\chi_{696}(53,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{17} a^{19} - \frac{5}{17} a^{18} - \frac{2}{17} a^{17} - \frac{1}{17} a^{16} + \frac{3}{17} a^{15} - \frac{7}{17} a^{14} + \frac{4}{17} a^{13} + \frac{2}{17} a^{12} - \frac{7}{17} a^{11} + \frac{2}{17} a^{10} + \frac{4}{17} a^{9} - \frac{2}{17} a^{8} + \frac{8}{17} a^{7} - \frac{7}{17} a^{6} - \frac{6}{17} a^{5} + \frac{1}{17} a^{4} - \frac{4}{17} a^{3} - \frac{5}{17} a^{2} + \frac{8}{17} a$, $\frac{1}{17} a^{20} + \frac{7}{17} a^{18} + \frac{6}{17} a^{17} - \frac{2}{17} a^{16} + \frac{8}{17} a^{15} + \frac{3}{17} a^{14} + \frac{5}{17} a^{13} + \frac{3}{17} a^{12} + \frac{1}{17} a^{11} - \frac{3}{17} a^{10} + \frac{1}{17} a^{9} - \frac{2}{17} a^{8} - \frac{1}{17} a^{7} - \frac{7}{17} a^{6} + \frac{5}{17} a^{5} + \frac{1}{17} a^{4} - \frac{8}{17} a^{3} + \frac{6}{17} a$, $\frac{1}{17} a^{21} + \frac{7}{17} a^{18} - \frac{5}{17} a^{17} - \frac{2}{17} a^{16} - \frac{1}{17} a^{15} + \frac{3}{17} a^{14} - \frac{8}{17} a^{13} + \frac{4}{17} a^{12} - \frac{5}{17} a^{11} + \frac{4}{17} a^{10} + \frac{4}{17} a^{9} - \frac{4}{17} a^{8} + \frac{5}{17} a^{7} + \frac{3}{17} a^{6} - \frac{8}{17} a^{5} + \frac{2}{17} a^{4} - \frac{6}{17} a^{3} + \frac{7}{17} a^{2} - \frac{5}{17} a$, $\frac{1}{17} a^{22} - \frac{4}{17} a^{18} - \frac{5}{17} a^{17} + \frac{6}{17} a^{16} - \frac{1}{17} a^{15} + \frac{7}{17} a^{14} - \frac{7}{17} a^{13} - \frac{2}{17} a^{12} + \frac{2}{17} a^{11} + \frac{7}{17} a^{10} + \frac{2}{17} a^{9} + \frac{2}{17} a^{8} - \frac{2}{17} a^{7} + \frac{7}{17} a^{6} - \frac{7}{17} a^{5} + \frac{4}{17} a^{4} + \frac{1}{17} a^{3} - \frac{4}{17} a^{2} - \frac{5}{17} a$, $\frac{1}{17} a^{23} - \frac{8}{17} a^{18} - \frac{2}{17} a^{17} - \frac{5}{17} a^{16} + \frac{2}{17} a^{15} - \frac{1}{17} a^{14} - \frac{3}{17} a^{13} - \frac{7}{17} a^{12} - \frac{4}{17} a^{11} - \frac{7}{17} a^{10} + \frac{1}{17} a^{9} + \frac{7}{17} a^{8} + \frac{5}{17} a^{7} - \frac{1}{17} a^{6} - \frac{3}{17} a^{5} + \frac{5}{17} a^{4} - \frac{3}{17} a^{3} - \frac{8}{17} a^{2} - \frac{2}{17} a$, $\frac{1}{17} a^{24} - \frac{8}{17} a^{18} - \frac{4}{17} a^{17} - \frac{6}{17} a^{16} + \frac{6}{17} a^{15} - \frac{8}{17} a^{14} + \frac{8}{17} a^{13} - \frac{5}{17} a^{12} + \frac{5}{17} a^{11} + \frac{5}{17} a^{9} + \frac{6}{17} a^{8} - \frac{5}{17} a^{7} - \frac{8}{17} a^{6} + \frac{8}{17} a^{5} + \frac{5}{17} a^{4} - \frac{6}{17} a^{3} - \frac{8}{17} a^{2} - \frac{4}{17} a$, $\frac{1}{17} a^{25} + \frac{7}{17} a^{18} - \frac{5}{17} a^{17} - \frac{2}{17} a^{16} - \frac{1}{17} a^{15} + \frac{3}{17} a^{14} - \frac{7}{17} a^{13} + \frac{4}{17} a^{12} - \frac{5}{17} a^{11} + \frac{4}{17} a^{10} + \frac{4}{17} a^{9} - \frac{4}{17} a^{8} + \frac{5}{17} a^{7} + \frac{3}{17} a^{6} + \frac{8}{17} a^{5} + \frac{2}{17} a^{4} - \frac{6}{17} a^{3} + \frac{7}{17} a^{2} - \frac{4}{17} a$, $\frac{1}{12794311985681} a^{26} + \frac{208169850339}{12794311985681} a^{25} + \frac{336266886752}{12794311985681} a^{24} + \frac{345774458726}{12794311985681} a^{23} + \frac{41697384031}{12794311985681} a^{22} + \frac{90901521674}{12794311985681} a^{21} - \frac{319999935836}{12794311985681} a^{20} + \frac{176365628489}{12794311985681} a^{19} + \frac{4093448677314}{12794311985681} a^{18} + \frac{1080536237184}{12794311985681} a^{17} + \frac{6101674016161}{12794311985681} a^{16} + \frac{77358825390}{12794311985681} a^{15} + \frac{407997431735}{12794311985681} a^{14} + \frac{1939893048266}{12794311985681} a^{13} + \frac{3108196904325}{12794311985681} a^{12} - \frac{2634021602818}{12794311985681} a^{11} - \frac{3432638849918}{12794311985681} a^{10} - \frac{965808187177}{12794311985681} a^{9} + \frac{326431026437}{12794311985681} a^{8} + \frac{3088108033936}{12794311985681} a^{7} - \frac{2420605897220}{12794311985681} a^{6} - \frac{121504169330}{12794311985681} a^{5} + \frac{1123145992962}{12794311985681} a^{4} + \frac{514140848819}{12794311985681} a^{3} - \frac{5868393342724}{12794311985681} a^{2} + \frac{291270247308}{752606587393} a - \frac{16294873710}{44270975729}$, $\frac{1}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{27} + \frac{150986452498106474974704725254475961647111218313373157898840209326565650}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{26} + \frac{632169836971241302085997981526898771949008915055157631539013266433614206397666117311}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{25} + \frac{1553193131276808915657305413017387573687385654291679177839415841737606790624806191641}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{24} - \frac{7549992868870340699900975766173339244787133863781846038040291033385907487687837432249}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{23} - \frac{202660035667323523624567214819026842119405726468802682115951480248996806354086903441}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{22} - \frac{4112137588718318048668070870804741741808761234269494914162484490554230142633080893026}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{21} - \frac{9553044381231467408676723676275330972305646903618483534480832269083243736494814323164}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{20} - \frac{10557498838355025573536943961296963333989947200135759864919738569608801568600675520353}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{19} + \frac{36795843160879964999003892735486889410144479800549509687692162784610253959401635007737}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{18} - \frac{78850640183953764620837540017873746141789516710515970129145447457104193502522239742618}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{17} - \frac{14861288204206183008590792169664702876769321727529344724125668481558656387199969266598}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{16} - \frac{32849018823685606006505357450872831913286213251850572832925941865986835941067468478026}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{15} + \frac{84946368766208108277437223745101550564485324412017437300702648056424708986337838056282}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{14} - \frac{35348211504283026330732019644356389694106601514107694926680007723558551501253747842697}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{13} - \frac{174497490305805126079199183685943527365963046427597600507828266627055247718384076382523}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{12} - \frac{155145119832498355103966198150031900551605148564067427698328224007514913826731491897420}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{11} - \frac{73301629553027389022892903864680486109763483609625732886779474452677434218347481542777}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{10} - \frac{161867413423152920131986475648171404995466467189358652185540072247770253285616023803626}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{9} - \frac{194693410497111241193076599449632190650657225336944699938019470367053783376409916498850}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{8} + \frac{58637027671717541319151717541041447730723967837208094858461201913849219365016282259914}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{7} + \frac{155315003560104751273157976404754480812351030701972122641050461804102640630940808245585}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{6} - \frac{964314553639046509973502138579166545055098542955674884719857871730483568485393149549}{24160468210047200083774210216205802701438719898501577299708291491580259437958931162493} a^{5} - \frac{5883537940626428049423787125003726163729713570258033892827808222131629184780536266048}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{4} + \frac{91056504622327188037149895739862210192967069340856782229936228588018509499722209738766}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{3} + \frac{91485637989329558111262573737578175847920285174855730559811415552474327176070705273}{410727959570802401424161573675498645924458238274526814095040955356864410445301829762381} a^{2} + \frac{5608512024327028318812337170826469688068283079773391969562408053636037500104971079852}{24160468210047200083774210216205802701438719898501577299708291491580259437958931162493} a - \frac{252079317595369151116941699804209726948932940969511063303030658830955410027739354196}{1421204012355717651986718248012106041261101170500092782335781852445897613997584186029}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{67652301431821070599756249754120051924979743956690217792158907746509812}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{27} + \frac{131387473958409045953462833436389358794814639223181123651474811673321146}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{26} - \frac{2765465665686975479863755129361976526670140354540787440626770713003231332}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{25} + \frac{3221138550025580571819814229411088624875645053905612221559778114916664466}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{24} - \frac{70347235505176304133998927663405100663753104621086245803843241517715502658}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{23} + \frac{72610369666804403648907699333624879914752651892386462946950428785302806292}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{22} - \frac{979298140548514416448306277090829903412271511641028334062768906861252145668}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{21} + \frac{768555400036779865269435764729877667549802909473978903790014227578926537433}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{20} - \frac{9402702229509635433954830672337990536991859041911203071963977423019528579738}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{19} + \frac{6024817836930730055021122903035878149578645099356481158298173893726672474561}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{18} - \frac{3446895904167921350615232466429087299843887447754956773951780405355839216050}{32102387375771083764240525172096688424020824239559031887082848452343800701} a^{17} + \frac{19660765299552930489441963265165795636055564096562124137012425381335808196553}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{16} - \frac{252682437590928867655178404011547718596686495802436694700094213544729118919386}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{15} + \frac{42135636003296594798368398658211290953402015996384007124389880553159183898828}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{14} - \frac{776746814936357853683872075647352073977493475920919816006977722482079568146710}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{13} - \frac{20490981700191155165525603461881448555719542813666041511596160449123254640637}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{12} - \frac{1691333313371297409702763763841552452643656535134280446511956205046957105039664}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{11} - \frac{169956368278860457934916498962872494788870386544672547311259972057166466514204}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{10} - \frac{2539757727014939854928525968571401227736062296278871743424226523030726568709702}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{9} - \frac{580186847473208874754827851592526975100689399440861352152264589435476975850594}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{8} - \frac{2562315226797976186088080788509719912163284237449629879531926005493498965160254}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{7} - \frac{699239320524809027229251206337076620788518477929377597353061425641064115266205}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{6} - \frac{1493114030089390290273425794947615525570869111208966124351296419050277717004852}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{5} - \frac{752245805381436168769798249481327548269178521551337883962496021533679938547841}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{4} - \frac{367074556904335317206650657966563606338204935431656033169775722846910962133092}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{3} - \frac{76492283646669872035608645349608627985348441240486648203576319253011774633359}{545740585388108423992088927925643703208354012072503542080408423689844611917} a^{2} - \frac{11055768414729691817563170696786855750526548866793302379899256244064842700688}{545740585388108423992088927925643703208354012072503542080408423689844611917} a + \frac{23983248865277118160525992401024537351288375889234302212169152758279643070}{32102387375771083764240525172096688424020824239559031887082848452343800701} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{14}$ (as 28T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), 7.7.594823321.1, 14.14.742003380228915810271232.1, 14.0.1622761392560638877063184384.1, 14.0.773792930870360792667.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
3Data not computed
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$
29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$