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Properties

Label 16.0.111...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.117\times 10^{22}$
Root discriminant \(23.88\)
Ramified primes $3,5,17$
Class number $6$ (GRH)
Class group [6] (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 4*x^14 + 13*x^13 - 13*x^12 + 49*x^11 + 132*x^10 - 637*x^9 + 381*x^8 + 2548*x^7 + 2112*x^6 - 3136*x^5 - 3328*x^4 - 13312*x^3 - 16384*x^2 + 16384*x + 65536)
 
gp: K = bnfinit(y^16 - y^15 - 4*y^14 + 13*y^13 - 13*y^12 + 49*y^11 + 132*y^10 - 637*y^9 + 381*y^8 + 2548*y^7 + 2112*y^6 - 3136*y^5 - 3328*y^4 - 13312*y^3 - 16384*y^2 + 16384*y + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 4*x^14 + 13*x^13 - 13*x^12 + 49*x^11 + 132*x^10 - 637*x^9 + 381*x^8 + 2548*x^7 + 2112*x^6 - 3136*x^5 - 3328*x^4 - 13312*x^3 - 16384*x^2 + 16384*x + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - 4*x^14 + 13*x^13 - 13*x^12 + 49*x^11 + 132*x^10 - 637*x^9 + 381*x^8 + 2548*x^7 + 2112*x^6 - 3136*x^5 - 3328*x^4 - 13312*x^3 - 16384*x^2 + 16384*x + 65536)
 

\( x^{16} - x^{15} - 4 x^{14} + 13 x^{13} - 13 x^{12} + 49 x^{11} + 132 x^{10} - 637 x^{9} + 381 x^{8} + \cdots + 65536 \) Copy content Toggle raw display

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(11173814592383056640625\) \(\medspace = 3^{8}\cdot 5^{12}\cdot 17^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(23.88\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}5^{3/4}17^{1/2}\approx 23.87880512635518$
Ramified primes:   \(3\), \(5\), \(17\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(67,·)$, $\chi_{255}(137,·)$, $\chi_{255}(203,·)$, $\chi_{255}(16,·)$, $\chi_{255}(86,·)$, $\chi_{255}(152,·)$, $\chi_{255}(154,·)$, $\chi_{255}(101,·)$, $\chi_{255}(103,·)$, $\chi_{255}(169,·)$, $\chi_{255}(239,·)$, $\chi_{255}(52,·)$, $\chi_{255}(118,·)$, $\chi_{255}(188,·)$, $\chi_{255}(254,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{464}a^{10}-\frac{1}{16}a^{9}-\frac{1}{2}a^{8}+\frac{1}{16}a^{7}+\frac{3}{16}a^{6}-\frac{51}{464}a^{5}-\frac{1}{2}a^{4}-\frac{1}{16}a^{3}-\frac{3}{16}a^{2}-\frac{1}{2}a-\frac{6}{29}$, $\frac{1}{1856}a^{11}-\frac{1}{1856}a^{10}-\frac{1}{16}a^{9}-\frac{31}{64}a^{8}+\frac{15}{64}a^{7}+\frac{65}{1856}a^{6}-\frac{67}{464}a^{5}-\frac{17}{64}a^{4}-\frac{31}{64}a^{3}-\frac{3}{16}a^{2}-\frac{35}{116}a-\frac{13}{29}$, $\frac{1}{37120}a^{12}-\frac{1}{7424}a^{11}+\frac{1}{1280}a^{9}+\frac{79}{256}a^{8}-\frac{799}{7424}a^{7}-\frac{1}{580}a^{6}-\frac{61}{256}a^{5}-\frac{31}{256}a^{4}+\frac{1}{5}a^{3}-\frac{181}{464}a^{2}-\frac{13}{116}a-\frac{1}{5}$, $\frac{1}{333337600}a^{13}+\frac{691}{333337600}a^{12}+\frac{2003}{8333440}a^{11}-\frac{83571}{333337600}a^{10}+\frac{484611}{11494400}a^{9}+\frac{7349657}{66667520}a^{8}+\frac{300577}{41667200}a^{7}-\frac{31071229}{333337600}a^{6}-\frac{11763643}{66667520}a^{5}-\frac{417823}{1436800}a^{4}-\frac{2359949}{5208400}a^{3}-\frac{40157}{520840}a^{2}+\frac{269351}{1302100}a+\frac{6199}{325525}$, $\frac{1}{1333350400}a^{14}-\frac{1}{1333350400}a^{13}-\frac{733}{333337600}a^{12}-\frac{30211}{1333350400}a^{11}+\frac{44851}{1333350400}a^{10}+\frac{73158017}{1333350400}a^{9}+\frac{108074349}{333337600}a^{8}+\frac{309809299}{1333350400}a^{7}+\frac{569216573}{1333350400}a^{6}-\frac{56562839}{333337600}a^{5}+\frac{9440161}{83334400}a^{4}+\frac{3961379}{10416800}a^{3}+\frac{190861}{5208400}a^{2}-\frac{198599}{1302100}a-\frac{104832}{325525}$, $\frac{1}{5333401600}a^{15}-\frac{1}{5333401600}a^{14}-\frac{1}{1333350400}a^{13}-\frac{18483}{5333401600}a^{12}+\frac{437811}{5333401600}a^{11}+\frac{3751729}{5333401600}a^{10}-\frac{35986863}{1333350400}a^{9}+\frac{1782429379}{5333401600}a^{8}+\frac{2062447421}{5333401600}a^{7}-\frac{570676547}{1333350400}a^{6}+\frac{10694327}{41667200}a^{5}+\frac{16746493}{41667200}a^{4}-\frac{536557}{1302100}a^{3}+\frac{979683}{2604200}a^{2}+\frac{312751}{1302100}a-\frac{120538}{325525}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{6}$, which has order $6$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{44541}{5333401600} a^{15} - \frac{44541}{1066680320} a^{14} - \frac{441}{20833600} a^{13} + \frac{44541}{183910400} a^{12} - \frac{579033}{1066680320} a^{11} + \frac{4498641}{5333401600} a^{10} - \frac{44541}{83334400} a^{9} - \frac{13251653}{1066680320} a^{8} + \frac{4498641}{183910400} a^{7} + \frac{44541}{5208400} a^{6} - \frac{4498641}{66667520} a^{5} - \frac{8061921}{83334400} a^{4} - \frac{3346749}{20833600} a^{3} + \frac{400869}{1302100} a + \frac{44541}{65105} \)  (order $30$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{1}{1041680}a^{15}-\frac{1309}{83334400}a^{14}+\frac{1}{325525}a^{13}+\frac{13}{1041680}a^{12}+\frac{79}{5208400}a^{11}+\frac{969}{5208400}a^{10}-\frac{24557}{16666880}a^{9}-\frac{4209}{5208400}a^{8}+\frac{13621}{5208400}a^{7}+\frac{1569}{260420}a^{6}+\frac{404}{325525}a^{5}-\frac{13741701}{83334400}a^{4}-\frac{208}{13021}a^{3}-\frac{11584}{325525}a^{2}+\frac{812569}{1302100}a+\frac{1024}{13021}$, $\frac{7589}{1333350400}a^{15}+\frac{3953}{333337600}a^{14}-\frac{126881}{1333350400}a^{13}+\frac{78177}{1333350400}a^{12}+\frac{115297}{333337600}a^{11}-\frac{220741}{333337600}a^{10}+\frac{2951553}{1333350400}a^{9}-\frac{6069713}{1333350400}a^{8}-\frac{4899729}{333337600}a^{7}+\frac{75881117}{1333350400}a^{6}+\frac{559317}{20833600}a^{5}-\frac{11236469}{83334400}a^{4}-\frac{3888833}{20833600}a^{3}+\frac{18772}{325525}a^{2}-\frac{30356}{325525}a+\frac{204101}{325525}$, $\frac{701}{36782080}a^{15}-\frac{257521}{5333401600}a^{14}+\frac{1}{325525}a^{13}+\frac{280661}{1066680320}a^{12}-\frac{3413309}{5333401600}a^{11}+\frac{370389}{183910400}a^{10}-\frac{9113}{16666880}a^{9}-\frac{60854361}{5333401600}a^{8}+\frac{140685229}{5333401600}a^{7}+\frac{10137}{1041680}a^{6}+\frac{70801}{2873600}a^{5}-\frac{8722401}{83334400}a^{4}+\frac{95329}{1041680}a^{3}-\frac{2135149}{5208400}a^{2}+\frac{426469}{1302100}a-\frac{356}{2245}$, $\frac{14257}{1333350400}a^{15}+\frac{5861}{333337600}a^{14}-\frac{98657}{1333350400}a^{13}+\frac{71009}{1333350400}a^{12}+\frac{54689}{333337600}a^{11}-\frac{37451}{166668800}a^{10}+\frac{5030801}{1333350400}a^{9}-\frac{5140881}{1333350400}a^{8}-\frac{5728913}{333337600}a^{7}+\frac{69944989}{1333350400}a^{6}+\frac{26621339}{333337600}a^{5}+\frac{148577}{5208400}a^{4}+\frac{120929}{1302100}a^{3}-\frac{711351}{5208400}a^{2}-\frac{949353}{1302100}a-\frac{126544}{325525}$, $\frac{39359}{5333401600}a^{15}-\frac{141987}{5333401600}a^{14}-\frac{3097}{666675200}a^{13}+\frac{799411}{5333401600}a^{12}-\frac{61107}{183910400}a^{11}+\frac{3753011}{5333401600}a^{10}-\frac{736367}{666675200}a^{9}-\frac{23952707}{5333401600}a^{8}+\frac{104675479}{5333401600}a^{7}-\frac{505927}{22988800}a^{6}+\frac{894303}{83334400}a^{5}-\frac{5814771}{83334400}a^{4}+\frac{115111}{5208400}a^{3}-\frac{514299}{2604200}a^{2}+\frac{10251}{44900}a-\frac{1244}{65105}$, $\frac{6643}{1066680320}a^{15}-\frac{122083}{5333401600}a^{14}+\frac{9191}{666675200}a^{13}+\frac{415411}{5333401600}a^{12}-\frac{2351687}{5333401600}a^{11}+\frac{6327347}{5333401600}a^{10}-\frac{96727}{666675200}a^{9}-\frac{30213443}{5333401600}a^{8}+\frac{86961303}{5333401600}a^{7}-\frac{16522763}{666675200}a^{6}+\frac{839007}{83334400}a^{5}+\frac{644787}{20833600}a^{4}+\frac{246951}{5208400}a^{3}-\frac{1206647}{5208400}a^{2}-\frac{53183}{325525}a+\frac{160419}{325525}$, $\frac{76227}{2666700800}a^{15}+\frac{76457}{2666700800}a^{14}-\frac{35773}{333337600}a^{13}+\frac{613319}{2666700800}a^{12}+\frac{490933}{2666700800}a^{11}+\frac{2773799}{2666700800}a^{10}+\frac{2375057}{333337600}a^{9}-\frac{22579223}{2666700800}a^{8}-\frac{41524389}{2666700800}a^{7}+\frac{25949193}{333337600}a^{6}+\frac{16213259}{83334400}a^{5}+\frac{18910397}{83334400}a^{4}+\frac{483819}{2604200}a^{3}-\frac{1161069}{5208400}a^{2}-\frac{390107}{325525}a-\frac{530637}{325525}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 38248.3956886 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 38248.3956886 \cdot 6}{30\cdot\sqrt{11173814592383056640625}}\cr\approx \mathstrut & 0.175784847161 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 4*x^14 + 13*x^13 - 13*x^12 + 49*x^11 + 132*x^10 - 637*x^9 + 381*x^8 + 2548*x^7 + 2112*x^6 - 3136*x^5 - 3328*x^4 - 13312*x^3 - 16384*x^2 + 16384*x + 65536)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - x^15 - 4*x^14 + 13*x^13 - 13*x^12 + 49*x^11 + 132*x^10 - 637*x^9 + 381*x^8 + 2548*x^7 + 2112*x^6 - 3136*x^5 - 3328*x^4 - 13312*x^3 - 16384*x^2 + 16384*x + 65536, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - x^15 - 4*x^14 + 13*x^13 - 13*x^12 + 49*x^11 + 132*x^10 - 637*x^9 + 381*x^8 + 2548*x^7 + 2112*x^6 - 3136*x^5 - 3328*x^4 - 13312*x^3 - 16384*x^2 + 16384*x + 65536);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - 4*x^14 + 13*x^13 - 13*x^12 + 49*x^11 + 132*x^10 - 637*x^9 + 381*x^8 + 2548*x^7 + 2112*x^6 - 3136*x^5 - 3328*x^4 - 13312*x^3 - 16384*x^2 + 16384*x + 65536);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^2\times C_4$ (as 16T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-255}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-3}, \sqrt{85})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{-51})\), \(\Q(\sqrt{-15}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-15}, \sqrt{-51})\), 4.4.325125.1, 4.0.36125.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), 8.0.4228250625.1, 8.0.105706265625.2, \(\Q(\zeta_{15})\), 8.0.105706265625.3, 8.0.105706265625.1, 8.8.105706265625.1, 8.0.1305015625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ R R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ R ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(17\) Copy content Toggle raw display 17.8.4.1$x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$