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Properties

Label 32.0.30424831247...0336.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{124}\cdot 3^{16}\cdot 7^{16}$
Root discriminant $67.24$
Ramified primes $2, 3, 7$
Class number $1224$ (GRH)
Class group $[3, 408]$ (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![116092, 172032, 226720, 344192, -2770312, -1013184, 18813856, -43372256, 58308178, -73052800, 142388096, -322075968, 599463788, -878355456, 1041246888, -1028800288, 866628755, -632761344, 405338832, -229791536, 115997916, -52347824, 21161272, -7665216, 2483598, -716976, 183240, -40992, 7916, -1280, 168, -16, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 16*x^31 + 168*x^30 - 1280*x^29 + 7916*x^28 - 40992*x^27 + 183240*x^26 - 716976*x^25 + 2483598*x^24 - 7665216*x^23 + 21161272*x^22 - 52347824*x^21 + 115997916*x^20 - 229791536*x^19 + 405338832*x^18 - 632761344*x^17 + 866628755*x^16 - 1028800288*x^15 + 1041246888*x^14 - 878355456*x^13 + 599463788*x^12 - 322075968*x^11 + 142388096*x^10 - 73052800*x^9 + 58308178*x^8 - 43372256*x^7 + 18813856*x^6 - 1013184*x^5 - 2770312*x^4 + 344192*x^3 + 226720*x^2 + 172032*x + 116092)
 
gp: K = bnfinit(x^32 - 16*x^31 + 168*x^30 - 1280*x^29 + 7916*x^28 - 40992*x^27 + 183240*x^26 - 716976*x^25 + 2483598*x^24 - 7665216*x^23 + 21161272*x^22 - 52347824*x^21 + 115997916*x^20 - 229791536*x^19 + 405338832*x^18 - 632761344*x^17 + 866628755*x^16 - 1028800288*x^15 + 1041246888*x^14 - 878355456*x^13 + 599463788*x^12 - 322075968*x^11 + 142388096*x^10 - 73052800*x^9 + 58308178*x^8 - 43372256*x^7 + 18813856*x^6 - 1013184*x^5 - 2770312*x^4 + 344192*x^3 + 226720*x^2 + 172032*x + 116092, 1)
 

Normalized defining polynomial

\( x^{32} - 16 x^{31} + 168 x^{30} - 1280 x^{29} + 7916 x^{28} - 40992 x^{27} + 183240 x^{26} - 716976 x^{25} + 2483598 x^{24} - 7665216 x^{23} + 21161272 x^{22} - 52347824 x^{21} + 115997916 x^{20} - 229791536 x^{19} + 405338832 x^{18} - 632761344 x^{17} + 866628755 x^{16} - 1028800288 x^{15} + 1041246888 x^{14} - 878355456 x^{13} + 599463788 x^{12} - 322075968 x^{11} + 142388096 x^{10} - 73052800 x^{9} + 58308178 x^{8} - 43372256 x^{7} + 18813856 x^{6} - 1013184 x^{5} - 2770312 x^{4} + 344192 x^{3} + 226720 x^{2} + 172032 x + 116092 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(30424831247408100720538934941471930111515239368421944590336=2^{124}\cdot 3^{16}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.24$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(672=2^{5}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{672}(1,·)$, $\chi_{672}(517,·)$, $\chi_{672}(265,·)$, $\chi_{672}(13,·)$, $\chi_{672}(533,·)$, $\chi_{672}(281,·)$, $\chi_{672}(29,·)$, $\chi_{672}(545,·)$, $\chi_{672}(293,·)$, $\chi_{672}(41,·)$, $\chi_{672}(433,·)$, $\chi_{672}(181,·)$, $\chi_{672}(449,·)$, $\chi_{672}(197,·)$, $\chi_{672}(589,·)$, $\chi_{672}(461,·)$, $\chi_{672}(337,·)$, $\chi_{672}(85,·)$, $\chi_{672}(377,·)$, $\chi_{672}(601,·)$, $\chi_{672}(349,·)$, $\chi_{672}(421,·)$, $\chi_{672}(97,·)$, $\chi_{672}(209,·)$, $\chi_{672}(617,·)$, $\chi_{672}(365,·)$, $\chi_{672}(125,·)$, $\chi_{672}(113,·)$, $\chi_{672}(629,·)$, $\chi_{672}(169,·)$, $\chi_{672}(505,·)$, $\chi_{672}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{11}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{13}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{14}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{15}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{11}$, $\frac{1}{475106} a^{28} - \frac{7}{237553} a^{27} + \frac{10215}{475106} a^{26} + \frac{105577}{475106} a^{25} - \frac{57849}{475106} a^{24} - \frac{79345}{475106} a^{23} - \frac{65315}{475106} a^{22} + \frac{11189}{237553} a^{21} - \frac{43008}{237553} a^{20} + \frac{20284}{237553} a^{19} + \frac{74693}{475106} a^{18} - \frac{51407}{237553} a^{17} - \frac{45670}{237553} a^{16} - \frac{5011}{15326} a^{15} - \frac{228059}{475106} a^{14} - \frac{101145}{237553} a^{13} - \frac{86185}{475106} a^{12} - \frac{108076}{237553} a^{11} - \frac{85639}{237553} a^{10} + \frac{22997}{475106} a^{9} + \frac{189983}{475106} a^{8} + \frac{45650}{237553} a^{7} - \frac{10992}{237553} a^{6} - \frac{86554}{237553} a^{5} - \frac{83689}{237553} a^{4} + \frac{98393}{237553} a^{3} + \frac{82184}{237553} a^{2} + \frac{17695}{237553} a - \frac{75558}{237553}$, $\frac{1}{475106} a^{29} + \frac{10019}{475106} a^{27} + \frac{5517}{237553} a^{26} - \frac{5089}{475106} a^{25} + \frac{60981}{475106} a^{24} + \frac{5810}{237553} a^{23} + \frac{29090}{237553} a^{22} - \frac{10277}{475106} a^{21} + \frac{24109}{475106} a^{20} - \frac{35007}{237553} a^{19} - \frac{3662}{237553} a^{18} - \frac{52709}{237553} a^{17} - \frac{8783}{475106} a^{16} + \frac{105125}{237553} a^{15} - \frac{34687}{237553} a^{14} + \frac{876}{2449} a^{13} - \frac{234977}{475106} a^{12} - \frac{109217}{475106} a^{11} - \frac{118459}{237553} a^{10} + \frac{36835}{475106} a^{9} - \frac{49787}{237553} a^{8} - \frac{84551}{237553} a^{7} - \frac{2889}{237553} a^{6} - \frac{107680}{237553} a^{5} + \frac{114512}{237553} a^{4} + \frac{34368}{237553} a^{3} - \frac{19494}{237553} a^{2} - \frac{65381}{237553} a - \frac{107600}{237553}$, $\frac{1}{18804647883269937687860546072596128001483966501666} a^{30} - \frac{15}{18804647883269937687860546072596128001483966501666} a^{29} - \frac{3993230145370477845968595947109198604074599}{9402323941634968843930273036298064000741983250833} a^{28} + \frac{111810444070373379687120686519057560914089787}{18804647883269937687860546072596128001483966501666} a^{27} - \frac{1705887511941378806862751861720290883006950809714}{9402323941634968843930273036298064000741983250833} a^{26} - \frac{1332542654338556041965756668462060956421334733008}{9402323941634968843930273036298064000741983250833} a^{25} + \frac{1564497811801937800282649384104090674368683569233}{9402323941634968843930273036298064000741983250833} a^{24} + \frac{18025645814851309047487802745782764154638946659}{193862349312061213276912846109238433004989345378} a^{23} - \frac{230510435335221070092596855254354457084271321573}{18804647883269937687860546072596128001483966501666} a^{22} + \frac{1471305170759463186495404371834976768053404647231}{18804647883269937687860546072596128001483966501666} a^{21} + \frac{1209671311590652125893059717782363719933762943933}{9402323941634968843930273036298064000741983250833} a^{20} - \frac{3765967732754785500197908151326461592782274524527}{18804647883269937687860546072596128001483966501666} a^{19} - \frac{1067839355884540904257892263445110456646879654966}{9402323941634968843930273036298064000741983250833} a^{18} - \frac{4261011945824291791929505940369941063732167835743}{18804647883269937687860546072596128001483966501666} a^{17} - \frac{8884684364889177162874978152760182598768442543}{39013792288941779435395323802066655604738519713} a^{16} - \frac{7129981307583090865691797056434553521370282145255}{18804647883269937687860546072596128001483966501666} a^{15} + \frac{4122266728580041633530961315091181126645114940712}{9402323941634968843930273036298064000741983250833} a^{14} - \frac{961439247346705359587331472499666483643059665034}{9402323941634968843930273036298064000741983250833} a^{13} - \frac{3436781253679797557672623579701683794578279082429}{9402323941634968843930273036298064000741983250833} a^{12} + \frac{30295866211592254880929276056675444505195474236}{303300772310805446578395904396711741959418814543} a^{11} - \frac{1446720932365479082509205928948311369802873390818}{9402323941634968843930273036298064000741983250833} a^{10} - \frac{4321226654268055187871567217201474620515260799875}{18804647883269937687860546072596128001483966501666} a^{9} - \frac{3399424244987550591412115226516751660817640654390}{9402323941634968843930273036298064000741983250833} a^{8} - \frac{4186032416783125920709418060446457665309070395953}{9402323941634968843930273036298064000741983250833} a^{7} + \frac{3947208640711679904774505482002829327121426378973}{9402323941634968843930273036298064000741983250833} a^{6} - \frac{233349714134700423950038353322638740633809533087}{553077878919704049642957237429297882396587250049} a^{5} - \frac{4389172491782898733838165419867020795898810678582}{9402323941634968843930273036298064000741983250833} a^{4} - \frac{4172530137134203687782809328253910012275448837213}{9402323941634968843930273036298064000741983250833} a^{3} - \frac{1968819180354573550688673608293711272804481691156}{9402323941634968843930273036298064000741983250833} a^{2} + \frac{2429956047169243552738291993281570080960373128561}{9402323941634968843930273036298064000741983250833} a + \frac{1447298589509308914910294117668262630511858688512}{9402323941634968843930273036298064000741983250833}$, $\frac{1}{550973794789527998972227641637715330735224030035068088418} a^{31} + \frac{14649921}{550973794789527998972227641637715330735224030035068088418} a^{30} - \frac{127605565422094435196560783595693037291009237720477}{550973794789527998972227641637715330735224030035068088418} a^{29} - \frac{6249494560322433847268744360323121327741802833605}{550973794789527998972227641637715330735224030035068088418} a^{28} + \frac{49206634732714709318043980829474661541903143088535962717}{275486897394763999486113820818857665367612015017534044209} a^{27} + \frac{59540768820175321000402449101901663016084369445235999996}{275486897394763999486113820818857665367612015017534044209} a^{26} - \frac{865917044431851838344410215259009896269138584378111109}{3487175916389417715014098997707059055286228038196633471} a^{25} - \frac{59140695041167061525035490788866456929680882259312379123}{275486897394763999486113820818857665367612015017534044209} a^{24} + \frac{101242833380147413302753676218336273067067800186651409981}{550973794789527998972227641637715330735224030035068088418} a^{23} - \frac{3772120927790888091070315546016160450583998672421167461}{16205111611456705852124342401109274433388942059854943777} a^{22} - \frac{11662126444766547061066998512670595978756882661566765832}{275486897394763999486113820818857665367612015017534044209} a^{21} + \frac{624608060926278235918678868426561306852267914020757371}{5680142214325030917239460223069230213765196185928536994} a^{20} + \frac{135459116699171831022314167143346562371868202866657925}{4338376336925417314741949934155238824686803386102898334} a^{19} + \frac{89006000625819656630705127363833841344617482024378233721}{550973794789527998972227641637715330735224030035068088418} a^{18} - \frac{41603354515385286132495794947349437721332903162431867077}{275486897394763999486113820818857665367612015017534044209} a^{17} + \frac{95433819024443222608624780819441406178003655810905936523}{550973794789527998972227641637715330735224030035068088418} a^{16} + \frac{111741252249758885869573943258678945864788261134304780667}{275486897394763999486113820818857665367612015017534044209} a^{15} - \frac{200741452704790839019104272120693969522048618869523701097}{550973794789527998972227641637715330735224030035068088418} a^{14} - \frac{137114724539124146145967833277566010275636590367753080143}{550973794789527998972227641637715330735224030035068088418} a^{13} - \frac{76457759422858191971396314758929248197151077810005754371}{275486897394763999486113820818857665367612015017534044209} a^{12} + \frac{214222066350018651455104855994183058934184157460102177475}{550973794789527998972227641637715330735224030035068088418} a^{11} - \frac{197988618120949775672837346545463029183036320191215592325}{550973794789527998972227641637715330735224030035068088418} a^{10} + \frac{17158443454025482855869268616433571670033491885507109533}{275486897394763999486113820818857665367612015017534044209} a^{9} + \frac{120717637366344503805735766324385645187217237694745789351}{550973794789527998972227641637715330735224030035068088418} a^{8} - \frac{122798342907613753558731216660313311935838524389701839176}{275486897394763999486113820818857665367612015017534044209} a^{7} - \frac{50276295317164985114109078966114001313591169937716814235}{275486897394763999486113820818857665367612015017534044209} a^{6} - \frac{23551818427871907352635941271276036817711698296214259187}{275486897394763999486113820818857665367612015017534044209} a^{5} + \frac{27987245707784629044200823869283870446105437127676365299}{275486897394763999486113820818857665367612015017534044209} a^{4} + \frac{24479006413141016511698718126754495260834767648199439956}{275486897394763999486113820818857665367612015017534044209} a^{3} - \frac{93993113100971222217291204280705615219870710703003370909}{275486897394763999486113820818857665367612015017534044209} a^{2} - \frac{5673103525133155515653150155457127211949676309694985533}{16205111611456705852124342401109274433388942059854943777} a - \frac{92060898475198469234951739580195007874042628289355414436}{275486897394763999486113820818857665367612015017534044209}$

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

Class group and class number

$C_{3}\times C_{408}$, which has order $1224$ (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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{10167859424300221621924}{45216501728339446723605217553} a^{30} + \frac{152517891364503324328860}{45216501728339446723605217553} a^{29} - \frac{1480063064081544233316904}{45216501728339446723605217553} a^{28} + \frac{10400505581476894320183796}{45216501728339446723605217553} a^{27} - \frac{58518944567548837154435444}{45216501728339446723605217553} a^{26} + \frac{3456520474114099330988192}{572360781371385401564623007} a^{25} - \frac{1076755869970152568030948048}{45216501728339446723605217553} a^{24} + \frac{3627132980722589038116132236}{45216501728339446723605217553} a^{23} - \frac{10355796817867987887381750292}{45216501728339446723605217553} a^{22} + \frac{24631801445561893622463220544}{45216501728339446723605217553} a^{21} - \frac{45630409244479455516666350522}{45216501728339446723605217553} a^{20} + \frac{51604167714233168999726904096}{45216501728339446723605217553} a^{19} + \frac{38788251673818786929488191614}{45216501728339446723605217553} a^{18} - \frac{422587152669497844004936827220}{45216501728339446723605217553} a^{17} + \frac{2901671157403508877453387525185}{90433003456678893447210435106} a^{16} - \frac{3561409150441672803868742635976}{45216501728339446723605217553} a^{15} + \frac{225956044945704994622413718558}{1458596829946433765277587663} a^{14} - \frac{11413827692581865362932909625560}{45216501728339446723605217553} a^{13} + \frac{500124549217674477180679958699}{1458596829946433765277587663} a^{12} - \frac{17412527954081426663311330639752}{45216501728339446723605217553} a^{11} + \frac{15859108255445423325515926295844}{45216501728339446723605217553} a^{10} - \frac{11308006717639671081994035548376}{45216501728339446723605217553} a^{9} + \frac{12085934258138691493972180842973}{90433003456678893447210435106} a^{8} - \frac{2459139469535100417517822894088}{45216501728339446723605217553} a^{7} + \frac{1191522747154698949549195007352}{45216501728339446723605217553} a^{6} - \frac{65851610152848523987088299040}{2659794219314085101388542209} a^{5} + \frac{936564766788736737275012314622}{45216501728339446723605217553} a^{4} - \frac{14796575897319044449254612944}{1458596829946433765277587663} a^{3} + \frac{578103310383246550873095136}{466149502354014914676342449} a^{2} + \frac{48284444185196076772761625680}{45216501728339446723605217553} a + \frac{25615363815461348766145689851}{45216501728339446723605217553} \) (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:  \( 36503341566423.26 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-14}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\zeta_{16})^+\), 4.0.100352.5, 4.4.903168.2, 4.0.18432.2, 8.0.796594176.2, 8.0.10070523904.2, 8.0.815712436224.2, 8.8.815712436224.1, 8.0.815712436224.1, 8.0.339738624.2, 8.0.815712436224.5, 8.8.417644767346688.4, 8.0.173946175488.1, \(\Q(\zeta_{32})^+\), 8.0.5156108238848.1, 16.0.665386778610493267378176.2, 16.0.174427151692069147083584569344.6, 16.0.26585452170716224216367104.1, 16.16.174427151692069147083584569344.1, 16.0.174427151692069147083584569344.5, 16.0.174427151692069147083584569344.3, 16.0.30257271966902092038144.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$