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Properties

Label 32.0.15729269092...000.15
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $61.29$
Ramified primes $2, 3, 5, 7$
Class number $2560$ (GRH)
Class group $[2, 4, 4, 80]$ (GRH)
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![130321, 0, -1937848, 0, 22080969, 0, -85434344, 0, 234944832, 0, -270080808, 0, 207480103, 0, -108450824, 0, 42422151, 0, -12535456, 0, 2894263, 0, -517992, 0, 72552, 0, -7696, 0, 609, 0, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 32*x^30 + 609*x^28 - 7696*x^26 + 72552*x^24 - 517992*x^22 + 2894263*x^20 - 12535456*x^18 + 42422151*x^16 - 108450824*x^14 + 207480103*x^12 - 270080808*x^10 + 234944832*x^8 - 85434344*x^6 + 22080969*x^4 - 1937848*x^2 + 130321)
 
gp: K = bnfinit(x^32 - 32*x^30 + 609*x^28 - 7696*x^26 + 72552*x^24 - 517992*x^22 + 2894263*x^20 - 12535456*x^18 + 42422151*x^16 - 108450824*x^14 + 207480103*x^12 - 270080808*x^10 + 234944832*x^8 - 85434344*x^6 + 22080969*x^4 - 1937848*x^2 + 130321, 1)
 

Normalized defining polynomial

\( x^{32} - 32 x^{30} + 609 x^{28} - 7696 x^{26} + 72552 x^{24} - 517992 x^{22} + 2894263 x^{20} - 12535456 x^{18} + 42422151 x^{16} - 108450824 x^{14} + 207480103 x^{12} - 270080808 x^{10} + 234944832 x^{8} - 85434344 x^{6} + 22080969 x^{4} - 1937848 x^{2} + 130321 \)

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:  \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.29$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(643,·)$, $\chi_{840}(391,·)$, $\chi_{840}(659,·)$, $\chi_{840}(533,·)$, $\chi_{840}(407,·)$, $\chi_{840}(281,·)$, $\chi_{840}(671,·)$, $\chi_{840}(169,·)$, $\chi_{840}(559,·)$, $\chi_{840}(433,·)$, $\chi_{840}(307,·)$, $\chi_{840}(181,·)$, $\chi_{840}(629,·)$, $\chi_{840}(449,·)$, $\chi_{840}(197,·)$, $\chi_{840}(839,·)$, $\chi_{840}(713,·)$, $\chi_{840}(587,·)$, $\chi_{840}(461,·)$, $\chi_{840}(463,·)$, $\chi_{840}(211,·)$, $\chi_{840}(349,·)$, $\chi_{840}(97,·)$, $\chi_{840}(743,·)$, $\chi_{840}(491,·)$, $\chi_{840}(83,·)$, $\chi_{840}(757,·)$, $\chi_{840}(377,·)$, $\chi_{840}(379,·)$, $\chi_{840}(253,·)$, $\chi_{840}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{57} a^{17} + \frac{5}{57} a^{15} + \frac{25}{57} a^{13} + \frac{11}{57} a^{11} - \frac{7}{19} a^{9} + \frac{28}{57} a^{7} + \frac{7}{57} a^{5} + \frac{16}{57} a^{3} + \frac{4}{57} a$, $\frac{1}{627} a^{18} + \frac{43}{627} a^{16} - \frac{298}{627} a^{14} + \frac{49}{627} a^{12} - \frac{2}{627} a^{10} - \frac{29}{627} a^{8} + \frac{15}{209} a^{6} - \frac{39}{209} a^{4} - \frac{62}{209} a^{2} + \frac{2}{33}$, $\frac{1}{627} a^{19} - \frac{1}{627} a^{17} + \frac{109}{627} a^{15} + \frac{203}{627} a^{13} + \frac{47}{209} a^{11} + \frac{268}{627} a^{9} + \frac{67}{627} a^{7} + \frac{202}{627} a^{5} - \frac{263}{627} a^{3} - \frac{46}{209} a$, $\frac{1}{627} a^{20} - \frac{1}{11} a^{16} + \frac{2}{11} a^{14} - \frac{1}{33} a^{12} - \frac{8}{33} a^{10} + \frac{2}{33} a^{8} + \frac{2}{33} a^{6} + \frac{2}{33} a^{4} + \frac{94}{627} a^{2} - \frac{3}{11}$, $\frac{1}{627} a^{21} - \frac{2}{627} a^{17} - \frac{238}{627} a^{15} + \frac{34}{209} a^{13} - \frac{58}{209} a^{11} + \frac{137}{627} a^{9} - \frac{101}{209} a^{7} - \frac{68}{209} a^{5} - \frac{280}{627} a^{3} + \frac{49}{627} a$, $\frac{1}{627} a^{22} + \frac{1}{11} a^{16} - \frac{4}{33} a^{14} + \frac{7}{33} a^{12} - \frac{4}{33} a^{10} + \frac{14}{33} a^{8} + \frac{5}{33} a^{6} - \frac{305}{627} a^{4} - \frac{2}{11} a^{2} + \frac{5}{11}$, $\frac{1}{627} a^{23} + \frac{2}{627} a^{17} + \frac{92}{209} a^{15} + \frac{4}{209} a^{13} - \frac{18}{209} a^{11} + \frac{167}{627} a^{9} - \frac{191}{627} a^{7} - \frac{21}{209} a^{5} + \frac{260}{627} a^{3} + \frac{65}{627} a$, $\frac{1}{9405} a^{24} + \frac{7}{9405} a^{18} - \frac{23}{627} a^{16} - \frac{1}{627} a^{14} + \frac{203}{3135} a^{12} + \frac{36}{209} a^{10} - \frac{106}{627} a^{8} - \frac{47}{9405} a^{6} + \frac{257}{627} a^{4} + \frac{221}{627} a^{2} - \frac{1}{495}$, $\frac{1}{9405} a^{25} + \frac{7}{9405} a^{19} - \frac{1}{627} a^{17} + \frac{109}{627} a^{15} - \frac{182}{3135} a^{13} - \frac{277}{627} a^{11} + \frac{59}{627} a^{9} - \frac{212}{9405} a^{7} - \frac{72}{209} a^{5} - \frac{18}{209} a^{3} + \frac{1301}{9405} a$, $\frac{1}{9405} a^{26} + \frac{7}{9405} a^{20} - \frac{1}{11} a^{16} - \frac{1}{5} a^{14} + \frac{10}{33} a^{12} + \frac{14}{33} a^{10} - \frac{647}{9405} a^{8} + \frac{13}{33} a^{6} + \frac{13}{33} a^{4} - \frac{4624}{9405} a^{2} - \frac{3}{11}$, $\frac{1}{9405} a^{27} + \frac{7}{9405} a^{21} - \frac{2}{627} a^{17} + \frac{68}{285} a^{15} + \frac{311}{627} a^{13} + \frac{244}{627} a^{11} + \frac{838}{9405} a^{9} - \frac{94}{627} a^{7} + \frac{5}{627} a^{5} - \frac{829}{9405} a^{3} + \frac{49}{627} a$, $\frac{1}{24838605} a^{28} + \frac{1022}{24838605} a^{26} - \frac{1099}{24838605} a^{24} + \frac{10837}{24838605} a^{22} - \frac{16666}{24838605} a^{20} - \frac{4753}{24838605} a^{18} - \frac{658697}{8279535} a^{16} + \frac{708337}{2759845} a^{14} + \frac{809181}{2759845} a^{12} - \frac{3145787}{24838605} a^{10} - \frac{9346294}{24838605} a^{8} - \frac{8359777}{24838605} a^{6} - \frac{6477079}{24838605} a^{4} - \frac{6152513}{24838605} a^{2} - \frac{4079}{68805}$, $\frac{1}{24838605} a^{29} + \frac{1022}{24838605} a^{27} - \frac{1099}{24838605} a^{25} + \frac{10837}{24838605} a^{23} - \frac{16666}{24838605} a^{21} - \frac{4753}{24838605} a^{19} + \frac{22526}{2759845} a^{17} - \frac{2523149}{8279535} a^{15} + \frac{4025348}{8279535} a^{13} - \frac{4017317}{24838605} a^{11} - \frac{5424409}{24838605} a^{9} + \frac{2970113}{24838605} a^{7} + \frac{8774696}{24838605} a^{5} + \frac{3870082}{24838605} a^{3} + \frac{381199}{1307295} a$, $\frac{1}{58045155948090697236600441004762788405} a^{30} + \frac{465978145810547606899956554212}{58045155948090697236600441004762788405} a^{28} + \frac{1235735413185950137061375309039554}{58045155948090697236600441004762788405} a^{26} + \frac{814528956039492009068060727473794}{58045155948090697236600441004762788405} a^{24} + \frac{42257313990560724916605745200040384}{58045155948090697236600441004762788405} a^{22} + \frac{2886564273857092627260361528217}{30247606017764823989890797813841995} a^{20} + \frac{608073400383013758845433045360686}{1758944119639112037472740636507963285} a^{18} + \frac{1340560402177557065203583846758261541}{19348385316030232412200147001587596135} a^{16} + \frac{3308228343845191212265896996031190432}{19348385316030232412200147001587596135} a^{14} + \frac{695833666809250632247569350841731584}{3055008207794247222978970579198041495} a^{12} - \frac{14775467703690271008949331027889297284}{58045155948090697236600441004762788405} a^{10} + \frac{3688640663727967518454416111642032377}{58045155948090697236600441004762788405} a^{8} + \frac{19301065906425441389112782666103954227}{58045155948090697236600441004762788405} a^{6} + \frac{24718048630369122992985968186006455247}{58045155948090697236600441004762788405} a^{4} - \frac{19343917250348242890163002343784587006}{58045155948090697236600441004762788405} a^{2} - \frac{5010828045945115270382587680541366}{53596635224460477596122290863123535}$, $\frac{1}{1102857963013723247495408379090492979695} a^{31} + \frac{3832224029386080053483573843260}{220571592602744649499081675818098595939} a^{29} - \frac{8275033398573294807884815425204904}{220571592602744649499081675818098595939} a^{27} + \frac{1922107823947435788053303357474432}{122539773668191471943934264343388108855} a^{25} + \frac{104516917349416535875708956201963817}{220571592602744649499081675818098595939} a^{23} + \frac{738330283354523457153274162929085}{2183877154482620292070115602159392039} a^{21} - \frac{365034807868428390912109192429448074}{1102857963013723247495408379090492979695} a^{19} - \frac{139105129602782064330971889914514957}{24507954733638294388786852868677621771} a^{17} - \frac{35273904836123627479576068353934010430}{73523864200914883166360558606032865313} a^{15} + \frac{332911813049875734815679543751049959387}{1102857963013723247495408379090492979695} a^{13} - \frac{8477657650389409825555834636819133297}{220571592602744649499081675818098595939} a^{11} - \frac{34776130075384312541764557363327619591}{220571592602744649499081675818098595939} a^{9} + \frac{24484917976600207914323918737002949414}{1102857963013723247495408379090492979695} a^{7} + \frac{49305932859245560155848477787770287571}{220571592602744649499081675818098595939} a^{5} + \frac{89970059124245166630649596725440213702}{220571592602744649499081675818098595939} a^{3} - \frac{1306471861540443507765482087086918289}{3055008207794247222978970579198041495} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{80}$, which has order $2560$ (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:  \( \frac{15107012574914564302643691197032}{19348385316030232412200147001587596135} a^{30} - \frac{481815311086435861477405369444696}{19348385316030232412200147001587596135} a^{28} + \frac{481541612817606980053155489346232}{1018336069264749074326323526399347165} a^{26} - \frac{115302807211170529398960510648941288}{19348385316030232412200147001587596135} a^{24} + \frac{1084020992318958158664164236960627328}{19348385316030232412200147001587596135} a^{22} - \frac{76366910143067603400822044859335429}{191568171445843885269308386154332635} a^{20} + \frac{14310632044142924397977183317589765888}{6449461772010077470733382333862532045} a^{18} - \frac{61669619755094919615249712500237137848}{6449461772010077470733382333862532045} a^{16} + \frac{207435457938027656169528313613486595464}{6449461772010077470733382333862532045} a^{14} - \frac{1576890618359013076359494066301889643432}{19348385316030232412200147001587596135} a^{12} + \frac{2982230578442115310185649261785738653912}{19348385316030232412200147001587596135} a^{10} - \frac{3801116617328109923329564582742684469016}{19348385316030232412200147001587596135} a^{8} + \frac{3215044086314397694410512440186737938816}{19348385316030232412200147001587596135} a^{6} - \frac{1032348023452177488069346023641041614896}{19348385316030232412200147001587596135} a^{4} + \frac{297325040681838093515801064315092086888}{19348385316030232412200147001587596135} a^{2} - \frac{6198844719156417310344480287121544}{17865545074820159198707430287707845} \) (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:  \( 16173035148400.742 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3\times C_4$ (as 32T34):

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 $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-105}) \), \(\Q(\sqrt{210}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-2}, \sqrt{-105})\), \(\Q(\sqrt{6}, \sqrt{-70})\), \(\Q(\sqrt{-3}, \sqrt{35})\), \(\Q(\sqrt{6}, \sqrt{35})\), \(\Q(\sqrt{-3}, \sqrt{-70})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{35})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-10}, \sqrt{42})\), \(\Q(\sqrt{-14}, \sqrt{30})\), \(\Q(\sqrt{7}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{42})\), \(\Q(\sqrt{-10}, \sqrt{-21})\), \(\Q(\sqrt{7}, \sqrt{30})\), \(\Q(\sqrt{-14}, \sqrt{-15})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-14})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{5}, \sqrt{-14})\), \(\Q(\sqrt{-21}, \sqrt{30})\), \(\Q(\sqrt{-15}, \sqrt{42})\), \(\Q(\sqrt{7}, \sqrt{-10})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{-15}, \sqrt{-21})\), \(\Q(\sqrt{30}, \sqrt{35})\), \(\Q(\sqrt{-10}, \sqrt{-14})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\zeta_{20})^+\), 4.0.55125.1, 4.4.3528000.1, 4.0.8000.2, 4.4.72000.1, 4.0.392000.2, 4.4.6125.1, 4.0.18000.1, 8.0.7965941760000.66, 8.0.7965941760000.55, 8.0.7965941760000.10, 8.0.7965941760000.27, 8.0.7965941760000.30, 8.0.31116960000.9, 8.0.7965941760000.1, 8.8.7965941760000.9, 8.0.7965941760000.15, 8.0.497871360000.19, 8.0.7965941760000.48, 8.0.207360000.2, 8.0.12745506816.5, 8.0.98344960000.6, 8.0.7965941760000.34, 8.0.777924000000.6, 8.0.199148544000000.138, 8.0.199148544000000.120, 8.0.777924000000.2, 8.8.199148544000000.7, 8.0.12446784000000.11, 8.8.12446784000000.4, 8.0.199148544000000.220, 8.0.1024000000.1, 8.0.12446784000000.7, 8.0.82944000000.5, 8.0.153664000000.3, 8.8.82944000000.2, 8.0.12446784000000.9, 8.8.12446784000000.1, 8.0.82944000000.4, 8.0.2458624000000.4, 8.0.12446784000000.3, 8.0.199148544000000.17, 8.0.153664000000.1, 8.8.9604000000.1, 8.0.777924000000.3, 8.8.199148544000000.2, 8.0.2458624000000.3, 8.0.324000000.2, 8.0.3038765625.1, 8.0.12446784000000.17, 8.0.5184000000.3, 16.0.63456228123711897600000000.15, 16.0.39660142577319936000000000000.12, 16.0.39660142577319936000000000000.40, 16.0.39660142577319936000000000000.7, 16.0.39660142577319936000000000000.17, 16.0.605165749776000000000000.1, 16.0.39660142577319936000000000000.37, 16.16.39660142577319936000000000000.2, 16.0.39660142577319936000000000000.30, 16.0.39660142577319936000000000000.43, 16.0.154922431942656000000000000.11, 16.0.6879707136000000000000.8, 16.0.154922431942656000000000000.16, 16.0.6044831973376000000000000.3, 16.0.39660142577319936000000000000.1

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 R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$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}$