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Properties

Label 32.0.15313006310...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 3^{16}\cdot 5^{16}\cdot 13^{24}$
Root discriminant $53.03$
Ramified primes $2, 3, 5, 13$
Class number $512$ (GRH)
Class group $[4, 4, 4, 8]$ (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![6561, 0, -8748, 0, -81486, 0, -167940, 0, 1297957, 0, 460998, 0, -657960, 0, -41841, 0, 257994, 0, -2988, 0, -81535, 0, 486, 0, 16752, 0, 1035, 0, -54, 0, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 9*x^30 - 54*x^28 + 1035*x^26 + 16752*x^24 + 486*x^22 - 81535*x^20 - 2988*x^18 + 257994*x^16 - 41841*x^14 - 657960*x^12 + 460998*x^10 + 1297957*x^8 - 167940*x^6 - 81486*x^4 - 8748*x^2 + 6561)
 
gp: K = bnfinit(x^32 + 9*x^30 - 54*x^28 + 1035*x^26 + 16752*x^24 + 486*x^22 - 81535*x^20 - 2988*x^18 + 257994*x^16 - 41841*x^14 - 657960*x^12 + 460998*x^10 + 1297957*x^8 - 167940*x^6 - 81486*x^4 - 8748*x^2 + 6561, 1)
 

Normalized defining polynomial

\( x^{32} + 9 x^{30} - 54 x^{28} + 1035 x^{26} + 16752 x^{24} + 486 x^{22} - 81535 x^{20} - 2988 x^{18} + 257994 x^{16} - 41841 x^{14} - 657960 x^{12} + 460998 x^{10} + 1297957 x^{8} - 167940 x^{6} - 81486 x^{4} - 8748 x^{2} + 6561 \)

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:  \(15313006310778859037593306549361889116160000000000000000=2^{32}\cdot 3^{16}\cdot 5^{16}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.03$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(780=2^{2}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{780}(1,·)$, $\chi_{780}(131,·)$, $\chi_{780}(389,·)$, $\chi_{780}(391,·)$, $\chi_{780}(649,·)$, $\chi_{780}(779,·)$, $\chi_{780}(109,·)$, $\chi_{780}(259,·)$, $\chi_{780}(151,·)$, $\chi_{780}(281,·)$, $\chi_{780}(541,·)$, $\chi_{780}(671,·)$, $\chi_{780}(161,·)$, $\chi_{780}(421,·)$, $\chi_{780}(551,·)$, $\chi_{780}(181,·)$, $\chi_{780}(521,·)$, $\chi_{780}(31,·)$, $\chi_{780}(701,·)$, $\chi_{780}(311,·)$, $\chi_{780}(79,·)$, $\chi_{780}(209,·)$, $\chi_{780}(469,·)$, $\chi_{780}(599,·)$, $\chi_{780}(571,·)$, $\chi_{780}(229,·)$, $\chi_{780}(359,·)$, $\chi_{780}(619,·)$, $\chi_{780}(749,·)$, $\chi_{780}(239,·)$, $\chi_{780}(499,·)$, $\chi_{780}(629,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{2}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{4}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{5}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{6}$, $\frac{1}{27} a^{19} + \frac{1}{27} a^{15} + \frac{1}{9} a^{11} - \frac{2}{27} a^{7} + \frac{4}{27} a^{3}$, $\frac{1}{81} a^{20} + \frac{1}{27} a^{18} - \frac{2}{81} a^{16} + \frac{4}{27} a^{12} - \frac{38}{81} a^{8} - \frac{8}{27} a^{6} + \frac{37}{81} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{81} a^{21} - \frac{2}{81} a^{17} - \frac{1}{27} a^{15} + \frac{4}{27} a^{13} - \frac{1}{9} a^{11} - \frac{11}{81} a^{9} - \frac{2}{9} a^{7} + \frac{10}{81} a^{5} - \frac{13}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{81} a^{22} - \frac{2}{81} a^{18} - \frac{1}{27} a^{16} + \frac{1}{27} a^{14} - \frac{1}{9} a^{12} - \frac{11}{81} a^{10} - \frac{2}{9} a^{8} + \frac{10}{81} a^{6} - \frac{13}{27} a^{4} + \frac{2}{9} a^{2}$, $\frac{1}{81} a^{23} + \frac{1}{81} a^{19} - \frac{1}{27} a^{17} - \frac{1}{27} a^{15} - \frac{1}{9} a^{13} - \frac{2}{81} a^{11} + \frac{1}{9} a^{9} + \frac{4}{81} a^{7} + \frac{5}{27} a^{5} + \frac{7}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{39528} a^{24} + \frac{14}{4941} a^{22} + \frac{10}{1647} a^{20} + \frac{1831}{39528} a^{18} + \frac{52}{4941} a^{16} - \frac{25}{1647} a^{14} + \frac{4693}{39528} a^{12} - \frac{55}{4941} a^{10} + \frac{368}{1647} a^{8} - \frac{9389}{39528} a^{6} + \frac{2002}{4941} a^{4} - \frac{16}{549} a^{2} + \frac{9}{488}$, $\frac{1}{39528} a^{25} + \frac{14}{4941} a^{23} + \frac{10}{1647} a^{21} + \frac{367}{39528} a^{19} + \frac{52}{4941} a^{17} - \frac{86}{1647} a^{15} + \frac{4693}{39528} a^{13} - \frac{604}{4941} a^{11} - \frac{181}{1647} a^{9} - \frac{6461}{39528} a^{7} - \frac{1292}{4941} a^{5} - \frac{292}{1647} a^{3} - \frac{461}{1464} a$, $\frac{1}{909144} a^{26} - \frac{1}{113643} a^{24} - \frac{430}{113643} a^{22} - \frac{1019}{303048} a^{20} - \frac{1838}{37881} a^{18} + \frac{1615}{113643} a^{16} - \frac{47747}{909144} a^{14} - \frac{692}{4941} a^{12} + \frac{12401}{113643} a^{10} - \frac{2117}{101016} a^{8} - \frac{2794}{37881} a^{6} - \frac{11573}{113643} a^{4} - \frac{28967}{101016} a^{2} + \frac{231}{1403}$, $\frac{1}{909144} a^{27} - \frac{1}{113643} a^{25} - \frac{430}{113643} a^{23} - \frac{1019}{303048} a^{21} - \frac{145}{12627} a^{19} + \frac{1615}{113643} a^{17} - \frac{14075}{909144} a^{15} - \frac{692}{4941} a^{13} - \frac{12853}{113643} a^{11} - \frac{2117}{101016} a^{9} + \frac{7027}{37881} a^{7} - \frac{11573}{113643} a^{5} - \frac{143021}{303048} a^{3} + \frac{231}{1403} a$, $\frac{1}{337716206020152} a^{28} - \frac{922453}{6621886392552} a^{26} - \frac{22876081}{2207295464184} a^{24} + \frac{290213136733}{112572068673384} a^{22} - \frac{1712650792361}{337716206020152} a^{20} - \frac{96377665261}{6621886392552} a^{18} - \frac{4635182352473}{112572068673384} a^{16} - \frac{5068015229849}{112572068673384} a^{14} - \frac{660344520311}{112572068673384} a^{12} + \frac{8513823396769}{112572068673384} a^{10} + \frac{39236314420315}{337716206020152} a^{8} + \frac{6667113129751}{112572068673384} a^{6} - \frac{141167762703823}{337716206020152} a^{4} + \frac{1696166633995}{12508007630376} a^{2} - \frac{505831034233}{4169335876792}$, $\frac{1}{1013148618060456} a^{29} - \frac{922453}{19865659177656} a^{27} - \frac{22876081}{6621886392552} a^{25} + \frac{290213136733}{337716206020152} a^{23} - \frac{1960662223051}{337716206020152} a^{21} - \frac{113877572279}{6621886392552} a^{19} + \frac{31957147587293}{1013148618060456} a^{17} - \frac{5068015229849}{337716206020152} a^{15} - \frac{1926409780831}{37524022891128} a^{13} + \frac{46037846287897}{337716206020152} a^{11} - \frac{46681709427247}{337716206020152} a^{9} + \frac{38356615308781}{112572068673384} a^{7} - \frac{257909167253999}{1013148618060456} a^{5} - \frac{274796582533}{4169335876792} a^{3} - \frac{4675166911025}{12508007630376} a$, $\frac{1}{7270028621558369240836385736} a^{30} - \frac{44011887247}{201945239487732478912121826} a^{28} - \frac{13958620644597056875}{47516526938289995038146312} a^{26} - \frac{172588691882473637857}{29917813257441848727721752} a^{24} + \frac{1361604246634652285106883}{605835718463197436736365478} a^{22} + \frac{2518454170114916902478171}{807780957950929915648487304} a^{20} + \frac{188321121097863101569531607}{7270028621558369240836385736} a^{18} - \frac{1667149686575425275289015}{67315079829244159637373942} a^{16} + \frac{737538478080281028380833}{47516526938289995038146312} a^{14} - \frac{3695625733179345342132781}{29917813257441848727721752} a^{12} + \frac{94749127390120564697197489}{605835718463197436736365478} a^{10} - \frac{698770996048511197138987}{3828345772279288699755864} a^{8} - \frac{2268465969798517888204197413}{7270028621558369240836385736} a^{6} - \frac{332942653872505722525521}{957086443069822174938966} a^{4} - \frac{12884616728906068260335633}{29917813257441848727721752} a^{2} - \frac{293374328675542354172609}{4986302209573641454620292}$, $\frac{1}{21810085864675107722509157208} a^{31} - \frac{44011887247}{605835718463197436736365478} a^{29} + \frac{532034815334510759}{1979855289095416459922763} a^{27} + \frac{29103377586421674941719}{2423342873852789746945461912} a^{25} + \frac{9981900760762141485521365}{1817507155389592310209096434} a^{23} - \frac{1672210886182435041410}{11219179971540693272895657} a^{21} + \frac{988985409203014558686713}{357542391224182093811625528} a^{19} - \frac{41567082753709021127413}{67315079829244159637373942} a^{17} - \frac{840028101053809774183445}{17818697601858748139304867} a^{15} + \frac{38731041173379568301006467}{2423342873852789746945461912} a^{13} - \frac{136847373205575485601878789}{1817507155389592310209096434} a^{11} + \frac{25039088624768970988856}{478543221534911087469483} a^{9} - \frac{783391993152253946306446847}{21810085864675107722509157208} a^{7} - \frac{1190886856700360097922223}{2871259329209466524816898} a^{5} + \frac{554892206220041580801312}{1246575552393410363655073} a^{3} + \frac{1424457371107801168785101}{3739726657180231090965219} a$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{8}$, which has order $512$ (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:  \( \frac{1367645762110748577687761}{21810085864675107722509157208} a^{31} + \frac{1381519372606879953772115}{2423342873852789746945461912} a^{29} - \frac{19781767967991696979321}{5939565867286249379768289} a^{27} + \frac{19577542128860555195224216}{302917859231598718368182739} a^{25} + \frac{7678364801871102437426988503}{7270028621558369240836385736} a^{23} + \frac{4397810500245328283889932}{33657539914622079818686971} a^{21} - \frac{27483652350736842388046939447}{5452521466168776930627289302} a^{19} - \frac{1588538495920772833826991745}{2423342873852789746945461912} a^{17} + \frac{16555505472968769369725944}{1048158682462279302312051} a^{15} - \frac{342789380767086702464549339}{302917859231598718368182739} a^{13} - \frac{293251675671964968831332282677}{7270028621558369240836385736} a^{11} + \frac{109263706388780141211276578}{4390113901907227802437431} a^{9} + \frac{441681142725295572787271225699}{5452521466168776930627289302} a^{7} - \frac{8516201874508435755251927}{9972604419147282909240584} a^{5} - \frac{456495571123242955499106}{1246575552393410363655073} a^{3} - \frac{70095015600809590825515559}{29917813257441848727721752} a \) (order $12$)
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:  \( 8672607098800.753 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{195}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{39}) \), \(\Q(\sqrt{-39}) \), \(\Q(i, \sqrt{195})\), \(\Q(i, \sqrt{65})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{3}, \sqrt{-65})\), \(\Q(\sqrt{-3}, \sqrt{65})\), \(\Q(\sqrt{-3}, \sqrt{-65})\), \(\Q(\sqrt{3}, \sqrt{65})\), \(\Q(i, \sqrt{13})\), \(\Q(i, \sqrt{15})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{39})\), \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\sqrt{-13}, \sqrt{15})\), \(\Q(\sqrt{-5}, \sqrt{39})\), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{13}, \sqrt{15})\), \(\Q(\sqrt{-13}, \sqrt{-15})\), \(\Q(\sqrt{-5}, \sqrt{-39})\), \(\Q(\sqrt{5}, \sqrt{39})\), \(\Q(\sqrt{-5}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{-13})\), \(\Q(\sqrt{-15}, \sqrt{39})\), \(\Q(\sqrt{15}, \sqrt{-39})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-5}, \sqrt{-13})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), \(\Q(\sqrt{15}, \sqrt{39})\), \(\Q(\sqrt{3}, \sqrt{13})\), \(\Q(\sqrt{3}, \sqrt{-13})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{-13})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), 4.0.54925.1, 4.4.878800.1, 4.4.19773.1, 4.0.316368.2, 4.4.35152.1, 4.0.2197.1, 4.0.7909200.1, 4.4.494325.1, 8.0.370150560000.10, 8.0.370150560000.2, 8.0.370150560000.8, 8.0.4569760000.1, 8.0.370150560000.7, 8.0.592240896.1, 8.0.12960000.1, 8.0.370150560000.3, 8.0.370150560000.5, 8.0.1445900625.1, 8.0.370150560000.1, 8.0.370150560000.4, 8.0.370150560000.9, 8.8.370150560000.1, 8.0.370150560000.6, 8.0.772289440000.3, 8.0.100088711424.1, 8.0.1235663104.1, 8.0.62555444640000.62, 8.0.244357205625.2, 8.0.62555444640000.22, 8.0.62555444640000.40, 8.0.244357205625.1, 8.0.62555444640000.61, 8.8.62555444640000.2, 8.8.62555444640000.3, 8.0.62555444640000.66, 8.0.772289440000.1, 8.0.772289440000.2, 8.0.62555444640000.43, 8.0.62555444640000.68, 8.0.3016755625.1, 8.8.772289440000.1, 8.8.244357205625.1, 8.0.62555444640000.63, 8.0.62555444640000.41, 8.8.62555444640000.4, 8.8.100088711424.1, 8.0.100088711424.3, 8.0.244357205625.3, 8.0.62555444640000.70, 8.0.390971529.1, 8.0.100088711424.2, 16.0.137011437068313600000000.1, 16.0.3913183654108104729600000000.9, 16.0.3913183654108104729600000000.7, 16.0.596430979135513600000000.1, 16.0.3913183654108104729600000000.8, 16.0.3913183654108104729600000000.3, 16.0.10017750154516748107776.1, 16.0.3913183654108104729600000000.1, 16.0.3913183654108104729600000000.5, 16.0.59710443940858531640625.1, 16.0.3913183654108104729600000000.6, 16.0.3913183654108104729600000000.10, 16.0.3913183654108104729600000000.4, 16.0.3913183654108104729600000000.2, 16.16.3913183654108104729600000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$