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Properties

Label 28.0.46614449603...0625.1
Degree $28$
Signature $[0, 14]$
Discriminant $3^{14}\cdot 5^{14}\cdot 43^{24}$
Root discriminant $97.31$
Ramified primes $3, 5, 43$
Class number $179249$ (GRH)
Class group $[179249]$ (GRH)
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![1, -62, 3315, -34194, 330317, -526838, 4110870, -2116445, 48825618, 13166733, 70217616, 23786690, 71237549, 23886766, 33359623, 7143500, 9302966, 1397646, 1783360, 113321, 228116, 1090, 21852, -1097, 1361, -100, 59, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 5*x^27 + 59*x^26 - 100*x^25 + 1361*x^24 - 1097*x^23 + 21852*x^22 + 1090*x^21 + 228116*x^20 + 113321*x^19 + 1783360*x^18 + 1397646*x^17 + 9302966*x^16 + 7143500*x^15 + 33359623*x^14 + 23886766*x^13 + 71237549*x^12 + 23786690*x^11 + 70217616*x^10 + 13166733*x^9 + 48825618*x^8 - 2116445*x^7 + 4110870*x^6 - 526838*x^5 + 330317*x^4 - 34194*x^3 + 3315*x^2 - 62*x + 1)
 
gp: K = bnfinit(x^28 - 5*x^27 + 59*x^26 - 100*x^25 + 1361*x^24 - 1097*x^23 + 21852*x^22 + 1090*x^21 + 228116*x^20 + 113321*x^19 + 1783360*x^18 + 1397646*x^17 + 9302966*x^16 + 7143500*x^15 + 33359623*x^14 + 23886766*x^13 + 71237549*x^12 + 23786690*x^11 + 70217616*x^10 + 13166733*x^9 + 48825618*x^8 - 2116445*x^7 + 4110870*x^6 - 526838*x^5 + 330317*x^4 - 34194*x^3 + 3315*x^2 - 62*x + 1, 1)
 

Normalized defining polynomial

\( x^{28} - 5 x^{27} + 59 x^{26} - 100 x^{25} + 1361 x^{24} - 1097 x^{23} + 21852 x^{22} + 1090 x^{21} + 228116 x^{20} + 113321 x^{19} + 1783360 x^{18} + 1397646 x^{17} + 9302966 x^{16} + 7143500 x^{15} + 33359623 x^{14} + 23886766 x^{13} + 71237549 x^{12} + 23786690 x^{11} + 70217616 x^{10} + 13166733 x^{9} + 48825618 x^{8} - 2116445 x^{7} + 4110870 x^{6} - 526838 x^{5} + 330317 x^{4} - 34194 x^{3} + 3315 x^{2} - 62 x + 1 \)

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:  \(46614449603597195528164649707632663661509314996337890625=3^{14}\cdot 5^{14}\cdot 43^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.31$
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:  $3, 5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(645=3\cdot 5\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{645}(256,·)$, $\chi_{645}(1,·)$, $\chi_{645}(514,·)$, $\chi_{645}(259,·)$, $\chi_{645}(4,·)$, $\chi_{645}(391,·)$, $\chi_{645}(64,·)$, $\chi_{645}(236,·)$, $\chi_{645}(11,·)$, $\chi_{645}(269,·)$, $\chi_{645}(16,·)$, $\chi_{645}(274,·)$, $\chi_{645}(451,·)$, $\chi_{645}(484,·)$, $\chi_{645}(226,·)$, $\chi_{645}(379,·)$, $\chi_{645}(164,·)$, $\chi_{645}(551,·)$, $\chi_{645}(41,·)$, $\chi_{645}(299,·)$, $\chi_{645}(44,·)$, $\chi_{645}(494,·)$, $\chi_{645}(431,·)$, $\chi_{645}(176,·)$, $\chi_{645}(434,·)$, $\chi_{645}(121,·)$, $\chi_{645}(59,·)$, $\chi_{645}(446,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{7} a^{20} - \frac{1}{7} a^{19} - \frac{1}{7} a^{17} + \frac{1}{7} a^{14} + \frac{1}{7} a^{13} + \frac{2}{7} a^{11} + \frac{2}{7} a^{10} + \frac{2}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{1}{7} a^{5} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{21} - \frac{1}{7} a^{19} - \frac{1}{7} a^{18} - \frac{1}{7} a^{17} + \frac{1}{7} a^{15} + \frac{2}{7} a^{14} + \frac{1}{7} a^{13} + \frac{2}{7} a^{12} - \frac{3}{7} a^{11} + \frac{2}{7} a^{10} + \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{22} - \frac{2}{7} a^{19} - \frac{1}{7} a^{18} - \frac{1}{7} a^{17} + \frac{1}{7} a^{16} + \frac{2}{7} a^{15} + \frac{2}{7} a^{14} + \frac{3}{7} a^{13} - \frac{3}{7} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} + \frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{23} - \frac{3}{7} a^{19} - \frac{1}{7} a^{18} - \frac{1}{7} a^{17} + \frac{2}{7} a^{16} + \frac{2}{7} a^{15} - \frac{2}{7} a^{14} - \frac{1}{7} a^{13} - \frac{3}{7} a^{12} + \frac{1}{7} a^{11} - \frac{2}{7} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{7} - \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{2}{7}$, $\frac{1}{7} a^{24} + \frac{3}{7} a^{19} - \frac{1}{7} a^{18} - \frac{1}{7} a^{17} + \frac{2}{7} a^{16} - \frac{2}{7} a^{15} + \frac{2}{7} a^{14} + \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{2}{7} a^{10} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{553} a^{25} - \frac{31}{553} a^{24} - \frac{12}{553} a^{23} + \frac{3}{79} a^{22} - \frac{16}{553} a^{21} - \frac{5}{553} a^{20} - \frac{223}{553} a^{19} - \frac{110}{553} a^{18} + \frac{167}{553} a^{17} - \frac{172}{553} a^{16} + \frac{248}{553} a^{15} - \frac{197}{553} a^{14} - \frac{270}{553} a^{13} - \frac{100}{553} a^{12} - \frac{183}{553} a^{11} - \frac{11}{553} a^{10} + \frac{156}{553} a^{9} - \frac{76}{553} a^{8} + \frac{152}{553} a^{7} + \frac{23}{553} a^{6} + \frac{250}{553} a^{5} + \frac{23}{553} a^{4} - \frac{29}{553} a^{3} + \frac{248}{553} a^{2} - \frac{258}{553} a + \frac{167}{553}$, $\frac{1}{23182776913800847} a^{26} + \frac{431718848755}{3311825273400121} a^{25} - \frac{1021234293397567}{23182776913800847} a^{24} + \frac{718888173756031}{23182776913800847} a^{23} + \frac{700673467485969}{23182776913800847} a^{22} - \frac{757395925645298}{23182776913800847} a^{21} - \frac{67285519864431}{23182776913800847} a^{20} - \frac{7398568760238461}{23182776913800847} a^{19} + \frac{1200036552362908}{3311825273400121} a^{18} + \frac{2073703453772954}{23182776913800847} a^{17} + \frac{763490770553949}{23182776913800847} a^{16} - \frac{3415405166232321}{23182776913800847} a^{15} + \frac{3754819590692760}{23182776913800847} a^{14} - \frac{9082123228364276}{23182776913800847} a^{13} - \frac{958291702378035}{3311825273400121} a^{12} - \frac{2888701323545180}{23182776913800847} a^{11} + \frac{855522387871089}{3311825273400121} a^{10} + \frac{1087733004696034}{23182776913800847} a^{9} - \frac{6603673312390521}{23182776913800847} a^{8} - \frac{5057624164136949}{23182776913800847} a^{7} + \frac{8470300775052933}{23182776913800847} a^{6} - \frac{4781330367995213}{23182776913800847} a^{5} - \frac{486460510176034}{3311825273400121} a^{4} - \frac{321053503065260}{3311825273400121} a^{3} - \frac{3801875921650551}{23182776913800847} a^{2} - \frac{32902863606783}{23182776913800847} a + \frac{767570995283612}{3311825273400121}$, $\frac{1}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{27} + \frac{11460286903917564738329959351896067266281381957000286324810117472331202}{572739225756779630649032140609374365282808527493431764155870860823800324842194428243513} a^{26} - \frac{1707809693487556893352174872604729047560401295004854354132808892228171848082593427744}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{25} - \frac{19672887213015045917316817267051250478352055247820915192829032670491419388306415204219}{572739225756779630649032140609374365282808527493431764155870860823800324842194428243513} a^{24} + \frac{244698588530722804379167813385576523010024827933004832973894165136362688549233996088561}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{23} + \frac{202423181395343814374561215869518351630964587326809569044641340248326215880657121914199}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{22} - \frac{177451875371375571558946516388993748038464835997739333484046847156559768346875137915372}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{21} - \frac{108404521309937333230353694653083145684489398892171120150826868773195025464712322469353}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{20} + \frac{552410756163539437899585186823957517088236991554421612202740460630924316678029412954308}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{19} - \frac{996321446861191427011146075856327854479840207257620147294699582176382792764050560333714}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{18} - \frac{661297451160196289541580060214289392882476809218291244396929545125088101140256446374469}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{17} + \frac{1314910689233677263836525172436322347386436693747154729291510008392226352464155435388451}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{16} - \frac{1813381037841880590860551596032033526066929421721355887315010721860032242497897938442672}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{15} - \frac{1887227871761087677948655437585851928471180489387773089547381153052994766157817563571741}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{14} + \frac{916302292301486326814557200758421341541995602134001986939966987081209552560718779809286}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{13} + \frac{1139936620192118540047895953009398146823680849398366753651827007424053286362124832016361}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{12} - \frac{609334715088115895044161719985075799541524564087788517491202835532372834136367945648341}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{11} - \frac{863955986365591127892127577501886837429057117963371315352647645634171463532600454061287}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{10} - \frac{1132466987877024086734099304482307854179407605706080899758766968104051399122253633034824}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{9} - \frac{359585652564183436924097890036089674141752473927611780572448597140630583073503095144888}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{8} + \frac{1854641686098128461245525101693457518459915944189510817642099241484432778055731669426448}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{7} - \frac{1802051598676668954291329851579309237984957382186203284481790598361448652189447812550407}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{6} + \frac{146876297140882047619442648949430179733141540219162660076693106678474677866807065930431}{572739225756779630649032140609374365282808527493431764155870860823800324842194428243513} a^{5} + \frac{860386893466359681825032720595319894574999937296146069674606021808207747464474835951551}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{4} + \frac{793074136873702507923758335234713605574338463083434570387242333153975035693812893264292}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a^{3} + \frac{195445895763474408671709961834984612667772855242875164425464087107384408893115452347067}{572739225756779630649032140609374365282808527493431764155870860823800324842194428243513} a^{2} - \frac{886816714534582840561533073961684567267103913480842651211752102686170884103488813033228}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591} a + \frac{661318139581007811262718852068821482421499524057370071728493703018914561214758710261426}{4009174580297457414543224984265620556979659692454022349091096025766602273895360997704591}$

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

Class group and class number

$C_{179249}$, which has order $179249$ (assuming GRH)

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{251818712124778812976018206829629576136021391243557076290737565342703646}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{27} + \frac{1255449376858713962115186955928145558662846071276364904513264574696479615}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{26} - \frac{14839537159135888423842107172556861489091445679997627298858160164865267803}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{25} + \frac{24969130281509613341651295215289986386700599481251510507455410651299364071}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{24} - \frac{342387613032382347212203046337281866775002125974252831868529471909954753508}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{23} + \frac{271330518640613351084676212620735529007499330921716447203427518572263947518}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{22} - \frac{5499362143899358022583545335807655427914564037906055606836146641882561867606}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{21} - \frac{353623543144264054201674760871509557337410063628947896537241034149961154879}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{20} - \frac{57457761772613935244366550269187925912315116210197958393240422753017104316429}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{19} - \frac{29368244700150813991097998211867056796761680134094339386996040095494677128118}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{18} - \frac{449599885346249169003866382999594684152241527200447624415613745005302083026714}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{17} - \frac{358504840660742530172732334504555788602315256114949591391876414093446139129815}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{16} - \frac{2348563671252560494871114707999337523487179828554703691596165182709968971781587}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{15} - \frac{1833411580964975774309334037816631500673799561726733217725619044784277746076706}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{14} - \frac{8430834102418800678784779759450017509603000125776976976363455961245463084774360}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{13} - \frac{6139989351446267903372817431371722974755659030711846002622800824589848861940986}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{12} - \frac{18041147737044878607774958322237001688748407707952873405348318533935993097467859}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{11} - \frac{6260538018575697198752770281604310651434931836086762845653147123123174038610841}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{10} - \frac{17801156839704433185054177126909471206927064495744679906817446475290086572827457}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{9} - \frac{3582666941142490633588000712159586898372537841955700909575610316584485043812232}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{8} - \frac{12375022110850810079684144193707557687599169488276660182366871072208164133387776}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{7} + \frac{348570605503902292661689287237570080969659617862467108805827088084715995580446}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{6} - \frac{1049566633547818027777429269133643117696820645621224647744287251251983350316709}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{5} + \frac{118238230168430427920554373805115096280466202173614777391276586812492397523820}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{4} - \frac{83011422365320835600954590539802480781605227923965383176029417901502674147158}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{3} + \frac{7448321797883190648559284636563171244064800289067386080033402417869027258329}{13662073056267515381064828472414865415555624726656156913462689746406987087} a^{2} - \frac{847756244906860237165707248949281151117281076604974216656993059924761883173}{13662073056267515381064828472414865415555624726656156913462689746406987087} a + \frac{15863827984898677314292828696037929585742435000355767600945377044352431503}{13662073056267515381064828472414865415555624726656156913462689746406987087} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 263819853122.8475 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 7.7.6321363049.1, 14.14.3121846156036138781328125.1, 14.0.87391712553613254588987.1, 14.0.6827477543251035514764609375.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

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
3Data not computed
5Data not computed
$43$43.14.12.1$x^{14} + 3569 x^{7} + 4043763$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$
43.14.12.1$x^{14} + 3569 x^{7} + 4043763$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$