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Properties

Label 32.0.73328647998...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{16}$
Root discriminant $41.76$
Ramified primes $2, 3, 5, 13$
Class number Not computed
Class group Not computed
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![43046721, 0, 33480783, 0, 4782969, 0, -9093546, 0, -7072758, 0, -4077297, 0, -523422, 0, 1107414, 0, 854761, 0, 123046, 0, -6462, 0, -5593, 0, -1078, 0, -154, 0, 9, 0, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 7*x^30 + 9*x^28 - 154*x^26 - 1078*x^24 - 5593*x^22 - 6462*x^20 + 123046*x^18 + 854761*x^16 + 1107414*x^14 - 523422*x^12 - 4077297*x^10 - 7072758*x^8 - 9093546*x^6 + 4782969*x^4 + 33480783*x^2 + 43046721)
 
gp: K = bnfinit(x^32 + 7*x^30 + 9*x^28 - 154*x^26 - 1078*x^24 - 5593*x^22 - 6462*x^20 + 123046*x^18 + 854761*x^16 + 1107414*x^14 - 523422*x^12 - 4077297*x^10 - 7072758*x^8 - 9093546*x^6 + 4782969*x^4 + 33480783*x^2 + 43046721, 1)
 

Normalized defining polynomial

\( x^{32} + 7 x^{30} + 9 x^{28} - 154 x^{26} - 1078 x^{24} - 5593 x^{22} - 6462 x^{20} + 123046 x^{18} + 854761 x^{16} + 1107414 x^{14} - 523422 x^{12} - 4077297 x^{10} - 7072758 x^{8} - 9093546 x^{6} + 4782969 x^{4} + 33480783 x^{2} + 43046721 \)

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:  \(7332864799814241410137948416000000000000000000000000=2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.76$
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$$\chi_{780}(1,·)$, $\chi_{780}(131,·)$, $\chi_{780}(389,·)$, $\chi_{780}(391,·)$, $\chi_{780}(521,·)$, $\chi_{780}(779,·)$, $\chi_{780}(259,·)$, $\chi_{780}(157,·)$, $\chi_{780}(287,·)$, $\chi_{780}(547,·)$, $\chi_{780}(677,·)$, $\chi_{780}(181,·)$, $\chi_{780}(649,·)$, $\chi_{780}(313,·)$, $\chi_{780}(443,·)$, $\chi_{780}(701,·)$, $\chi_{780}(53,·)$, $\chi_{780}(599,·)$, $\chi_{780}(311,·)$, $\chi_{780}(77,·)$, $\chi_{780}(79,·)$, $\chi_{780}(209,·)$, $\chi_{780}(467,·)$, $\chi_{780}(469,·)$, $\chi_{780}(727,·)$, $\chi_{780}(571,·)$, $\chi_{780}(337,·)$, $\chi_{780}(233,·)$, $\chi_{780}(103,·)$, $\chi_{780}(493,·)$, $\chi_{780}(623,·)$, $\chi_{780}(703,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{18} - \frac{2}{9} a^{16} - \frac{1}{9} a^{12} + \frac{2}{9} a^{10} - \frac{4}{9} a^{8} - \frac{2}{9} a^{4} + \frac{4}{9} a^{2}$, $\frac{1}{27} a^{19} - \frac{2}{27} a^{17} + \frac{8}{27} a^{13} + \frac{11}{27} a^{11} + \frac{5}{27} a^{9} + \frac{7}{27} a^{5} + \frac{13}{27} a^{3}$, $\frac{1}{93879} a^{20} + \frac{1}{81} a^{18} - \frac{1}{3} a^{16} + \frac{23}{81} a^{14} + \frac{8}{81} a^{12} - \frac{27022}{93879} a^{10} + \frac{1}{3} a^{8} + \frac{10}{81} a^{6} + \frac{7}{81} a^{4} - \frac{1}{3} a^{2} - \frac{430}{1159}$, $\frac{1}{281637} a^{21} + \frac{1}{243} a^{19} - \frac{1}{9} a^{17} + \frac{104}{243} a^{15} + \frac{89}{243} a^{13} - \frac{120901}{281637} a^{11} - \frac{2}{9} a^{9} + \frac{10}{243} a^{7} + \frac{88}{243} a^{5} + \frac{2}{9} a^{3} - \frac{430}{3477} a$, $\frac{1}{844911} a^{22} - \frac{2}{844911} a^{20} + \frac{1}{27} a^{18} + \frac{347}{729} a^{16} + \frac{116}{729} a^{14} + \frac{98150}{844911} a^{12} + \frac{1787}{31293} a^{10} - \frac{233}{729} a^{8} - \frac{101}{729} a^{6} + \frac{7}{27} a^{4} - \frac{430}{10431} a^{2} - \frac{162}{1159}$, $\frac{1}{2534733} a^{23} - \frac{2}{2534733} a^{21} + \frac{1}{81} a^{19} + \frac{347}{2187} a^{17} + \frac{845}{2187} a^{15} - \frac{746761}{2534733} a^{13} + \frac{1787}{93879} a^{11} + \frac{496}{2187} a^{9} + \frac{628}{2187} a^{7} + \frac{7}{81} a^{5} + \frac{10001}{31293} a^{3} + \frac{997}{3477} a$, $\frac{1}{304167960} a^{24} + \frac{4}{7604199} a^{22} + \frac{1}{281637} a^{20} - \frac{6943}{262440} a^{18} - \frac{961}{6561} a^{16} + \frac{3546386}{7604199} a^{14} - \frac{35173}{11265480} a^{12} + \frac{1813252}{7604199} a^{10} + \frac{202}{6561} a^{8} - \frac{4541}{9720} a^{6} - \frac{39127}{93879} a^{4} - \frac{835}{10431} a^{2} + \frac{8561}{46360}$, $\frac{1}{912503880} a^{25} + \frac{4}{22812597} a^{23} + \frac{1}{844911} a^{21} - \frac{6943}{787320} a^{19} - \frac{961}{19683} a^{17} + \frac{11150585}{22812597} a^{15} + \frac{11230307}{33796440} a^{13} + \frac{9417451}{22812597} a^{11} - \frac{6359}{19683} a^{9} - \frac{4541}{29160} a^{7} - \frac{39127}{281637} a^{5} - \frac{11266}{31293} a^{3} + \frac{18307}{46360} a$, $\frac{1}{1207228945681800} a^{26} + \frac{20347}{63538365562200} a^{24} + \frac{1418701}{3353413738005} a^{22} + \frac{187350677}{63538365562200} a^{20} - \frac{16872007159}{1041612550200} a^{18} + \frac{1476966419279}{30180723642045} a^{16} - \frac{1577962199461}{7059818395800} a^{14} + \frac{300390164806777}{1207228945681800} a^{12} + \frac{122948617627}{1588459139055} a^{10} + \frac{40045922737}{115734727800} a^{8} + \frac{1537258154837}{4968020352600} a^{6} + \frac{727128557}{2178956295} a^{4} - \frac{26953615199}{184000753800} a^{2} + \frac{100433363}{1076027800}$, $\frac{1}{3621686837045400} a^{27} + \frac{20347}{190615096686600} a^{25} + \frac{1418701}{10060241214015} a^{23} + \frac{187350677}{190615096686600} a^{21} - \frac{16872007159}{3124837650600} a^{19} + \frac{1476966419279}{90542170926135} a^{17} + \frac{5481856196339}{21179455187400} a^{15} + \frac{300390164806777}{3621686837045400} a^{13} - \frac{1465510521428}{4765377417165} a^{11} + \frac{155780650537}{347204183400} a^{9} + \frac{1537258154837}{14904061057800} a^{7} + \frac{727128557}{6536868885} a^{5} - \frac{210954368999}{552002261400} a^{3} + \frac{100433363}{3228083400} a$, $\frac{1}{10865060511136200} a^{28} - \frac{1}{5432530255568100} a^{26} + \frac{15841}{16096385942424} a^{24} + \frac{4715046863}{10865060511136200} a^{22} + \frac{16288810417}{5432530255568100} a^{20} + \frac{107968749843601}{2173012102227240} a^{18} - \frac{118133288873653}{402409648560600} a^{16} + \frac{281231401734611}{5432530255568100} a^{14} - \frac{898026526144369}{2173012102227240} a^{12} - \frac{194254622736539}{402409648560600} a^{10} + \frac{7460180074273}{67068274760100} a^{8} + \frac{145893023177}{331201356840} a^{6} - \frac{563677858199}{1656006784200} a^{4} + \frac{7647941771}{92000376900} a^{2} - \frac{250683297}{4088905640}$, $\frac{1}{32595181533408600} a^{29} - \frac{1}{16297590766704300} a^{27} + \frac{15841}{48289157827272} a^{25} + \frac{4715046863}{32595181533408600} a^{23} + \frac{16288810417}{16297590766704300} a^{21} + \frac{107968749843601}{6519036306681720} a^{19} - \frac{118133288873653}{1207228945681800} a^{17} + \frac{281231401734611}{16297590766704300} a^{15} - \frac{898026526144369}{6519036306681720} a^{13} - \frac{194254622736539}{1207228945681800} a^{11} - \frac{59608094685827}{201204824280300} a^{9} + \frac{145893023177}{993604070520} a^{7} - \frac{563677858199}{4968020352600} a^{5} + \frac{7647941771}{276001130700} a^{3} - \frac{83561099}{4088905640} a$, $\frac{1}{97785544600225800} a^{30} - \frac{1}{48892772300112900} a^{28} + \frac{1}{3621686837045400} a^{26} - \frac{18925892}{12223193075028225} a^{24} + \frac{5047951177}{48892772300112900} a^{22} + \frac{325060708181}{97785544600225800} a^{20} + \frac{12584425038397}{452710854630675} a^{18} - \frac{14232622804879429}{48892772300112900} a^{16} - \frac{21840126673880957}{97785544600225800} a^{14} - \frac{194335761049729}{452710854630675} a^{12} + \frac{41726743474831}{201204824280300} a^{10} + \frac{6531979654501}{134136549520200} a^{8} - \frac{362885467879}{1863007632225} a^{6} + \frac{169051836191}{828003392100} a^{4} + \frac{18837951307}{184000753800} a^{2} + \frac{4489051209}{20444528200}$, $\frac{1}{293356633800677400} a^{31} - \frac{1}{146678316900338700} a^{29} + \frac{1}{10865060511136200} a^{27} - \frac{18925892}{36669579225084675} a^{25} + \frac{5047951177}{146678316900338700} a^{23} + \frac{325060708181}{293356633800677400} a^{21} + \frac{12584425038397}{1358132563892025} a^{19} - \frac{14232622804879429}{146678316900338700} a^{17} - \frac{119625671274106757}{293356633800677400} a^{15} - \frac{647046615680404}{1358132563892025} a^{13} + \frac{242931567755131}{603614472840900} a^{11} + \frac{6531979654501}{402409648560600} a^{9} - \frac{362885467879}{5589022896675} a^{7} + \frac{169051836191}{2484010176300} a^{5} - \frac{165162802493}{552002261400} a^{3} - \frac{15955476991}{61333584600} a$

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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{418440841}{32595181533408600} a^{31} - \frac{59777263}{3621686837045400} a^{29} + \frac{4602849251}{16297590766704300} a^{27} + \frac{32219944757}{16297590766704300} a^{25} + \frac{9298923751}{3621686837045400} a^{23} + \frac{21460037417}{1810843418522700} a^{21} - \frac{3677676551549}{16297590766704300} a^{19} - \frac{51095273099143}{32595181533408600} a^{17} - \frac{3677676551549}{1810843418522700} a^{15} + \frac{540790624110293}{16297590766704300} a^{13} + \frac{334334231959}{44712183173400} a^{11} + \frac{32219944757}{2484010176300} a^{9} + \frac{4602849251}{276001130700} a^{7} - \frac{179331789}{20444528200} a^{5} - \frac{643198803}{10222264100} a^{3} - \frac{1613986101}{20444528200} a \) (order $60$)
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

$C_2^3\times C_4$ (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{195}) \), \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{39}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{15}) \), \(\Q(i, \sqrt{195})\), \(\Q(i, \sqrt{65})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{-65})\), \(\Q(\sqrt{3}, \sqrt{65})\), \(\Q(\sqrt{3}, \sqrt{-65})\), \(\Q(\sqrt{-3}, \sqrt{65})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{39})\), \(\Q(i, \sqrt{13})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{39})\), \(\Q(\sqrt{-5}, \sqrt{-39})\), \(\Q(\sqrt{-13}, \sqrt{-15})\), \(\Q(\sqrt{13}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{-5}, \sqrt{39})\), \(\Q(\sqrt{-13}, \sqrt{15})\), \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-13})\), \(\Q(\sqrt{-5}, \sqrt{13})\), \(\Q(\sqrt{-15}, \sqrt{39})\), \(\Q(\sqrt{15}, \sqrt{-39})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-5}, \sqrt{-13})\), \(\Q(\sqrt{15}, \sqrt{39})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{-13})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{13})\), \(\Q(\sqrt{3}, \sqrt{-13})\), 4.4.338000.1, 4.0.21125.1, \(\Q(\zeta_{15})^+\), 4.0.18000.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 4.0.3042000.1, 4.4.190125.1, 8.0.370150560000.10, 8.0.370150560000.8, 8.0.370150560000.2, 8.0.4569760000.1, 8.0.370150560000.7, 8.0.12960000.1, 8.0.592240896.1, 8.0.370150560000.9, 8.0.370150560000.4, 8.8.370150560000.1, 8.0.370150560000.6, 8.0.370150560000.5, 8.0.370150560000.3, 8.0.1445900625.1, 8.0.370150560000.1, 8.0.114244000000.2, 8.0.324000000.1, \(\Q(\zeta_{20})\), 8.0.9253764000000.8, 8.8.9253764000000.3, 8.0.9253764000000.6, 8.0.9253764000000.10, 8.8.9253764000000.2, 8.0.9253764000000.2, 8.0.36147515625.1, 8.0.36147515625.3, 8.0.9253764000000.5, 8.0.114244000000.3, 8.0.114244000000.1, 8.0.9253764000000.7, 8.0.9253764000000.1, 8.8.114244000000.1, 8.0.446265625.1, 8.8.36147515625.1, 8.0.9253764000000.3, 8.0.9253764000000.9, 8.0.36147515625.2, \(\Q(\zeta_{15})\), 8.0.324000000.2, 8.8.9253764000000.1, 8.0.9253764000000.4, \(\Q(\zeta_{60})^+\), 8.0.324000000.3, 16.0.137011437068313600000000.1, 16.0.85632148167696000000000000.8, 16.0.85632148167696000000000000.3, 16.0.13051691536000000000000.1, 16.0.85632148167696000000000000.5, 16.0.85632148167696000000000000.4, \(\Q(\zeta_{60})\), 16.0.85632148167696000000000000.2, 16.0.85632148167696000000000000.10, 16.16.85632148167696000000000000.1, 16.0.85632148167696000000000000.6, 16.0.85632148167696000000000000.9, 16.0.85632148167696000000000000.1, 16.0.85632148167696000000000000.7, 16.0.1306642885859619140625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$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$3.8.4.1$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}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$