Properties

Label 32.0.520...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $5.204\times 10^{47}$
Root discriminant \(30.98\)
Ramified primes $2,3,5$
Class number $20$ (GRH)
Class group [20] (GRH)
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 47*x^24 + 2208*x^16 - 47*x^8 + 1)
 
gp: K = bnfinit(y^32 - 47*y^24 + 2208*y^16 - 47*y^8 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 47*x^24 + 2208*x^16 - 47*x^8 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 47*x^24 + 2208*x^16 - 47*x^8 + 1)
 

\( x^{32} - 47x^{24} + 2208x^{16} - 47x^{8} + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(520402924666472696020370152488960000000000000000\) \(\medspace = 2^{96}\cdot 3^{16}\cdot 5^{16}\) 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:  \(30.98\)
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:  $2^{3}3^{1/2}5^{1/2}\approx 30.983866769659336$
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(240=2^{4}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{240}(1,·)$, $\chi_{240}(131,·)$, $\chi_{240}(11,·)$, $\chi_{240}(19,·)$, $\chi_{240}(149,·)$, $\chi_{240}(151,·)$, $\chi_{240}(29,·)$, $\chi_{240}(31,·)$, $\chi_{240}(161,·)$, $\chi_{240}(41,·)$, $\chi_{240}(71,·)$, $\chi_{240}(49,·)$, $\chi_{240}(179,·)$, $\chi_{240}(181,·)$, $\chi_{240}(59,·)$, $\chi_{240}(61,·)$, $\chi_{240}(191,·)$, $\chi_{240}(139,·)$, $\chi_{240}(199,·)$, $\chi_{240}(119,·)$, $\chi_{240}(79,·)$, $\chi_{240}(209,·)$, $\chi_{240}(211,·)$, $\chi_{240}(89,·)$, $\chi_{240}(91,·)$, $\chi_{240}(221,·)$, $\chi_{240}(101,·)$, $\chi_{240}(229,·)$, $\chi_{240}(109,·)$, $\chi_{240}(239,·)$, $\chi_{240}(169,·)$, $\chi_{240}(121,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{21}a^{16}+\frac{8}{21}a^{8}+\frac{1}{21}$, $\frac{1}{21}a^{17}+\frac{8}{21}a^{9}+\frac{1}{21}a$, $\frac{1}{21}a^{18}+\frac{8}{21}a^{10}+\frac{1}{21}a^{2}$, $\frac{1}{21}a^{19}+\frac{8}{21}a^{11}+\frac{1}{21}a^{3}$, $\frac{1}{21}a^{20}+\frac{8}{21}a^{12}+\frac{1}{21}a^{4}$, $\frac{1}{21}a^{21}+\frac{8}{21}a^{13}+\frac{1}{21}a^{5}$, $\frac{1}{21}a^{22}+\frac{8}{21}a^{14}+\frac{1}{21}a^{6}$, $\frac{1}{21}a^{23}+\frac{8}{21}a^{15}+\frac{1}{21}a^{7}$, $\frac{1}{46368}a^{24}-\frac{17711}{46368}$, $\frac{1}{46368}a^{25}-\frac{17711}{46368}a$, $\frac{1}{46368}a^{26}-\frac{17711}{46368}a^{2}$, $\frac{1}{46368}a^{27}-\frac{17711}{46368}a^{3}$, $\frac{1}{46368}a^{28}-\frac{17711}{46368}a^{4}$, $\frac{1}{46368}a^{29}-\frac{17711}{46368}a^{5}$, $\frac{1}{46368}a^{30}-\frac{17711}{46368}a^{6}$, $\frac{1}{46368}a^{31}-\frac{17711}{46368}a^{7}$ 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

$C_{20}$, which has order $20$ (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:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{4183}{46368} a^{27} + \frac{89}{21} a^{19} - \frac{4181}{21} a^{11} + \frac{89}{46368} a^{3} \)  (order $48$) 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:   $\frac{1}{5796}a^{31}+\frac{104005}{5796}a^{7}$, $\frac{2255}{15456}a^{28}-\frac{48}{7}a^{20}+\frac{2255}{7}a^{12}-\frac{105985}{15456}a^{4}-1$, $\frac{17711}{46368}a^{30}-\frac{377}{21}a^{22}+\frac{17711}{21}a^{14}-\frac{832417}{46368}a^{6}-1$, $\frac{28657}{46368}a^{31}-\frac{10951}{46368}a^{29}-\frac{610}{21}a^{23}+\frac{233}{21}a^{21}+\frac{28657}{21}a^{15}-\frac{10946}{21}a^{13}-\frac{1346879}{46368}a^{7}+\frac{233}{46368}a^{5}$, $\frac{4183}{46368}a^{27}-\frac{89}{21}a^{19}+\frac{4181}{21}a^{11}-\frac{89}{46368}a^{3}+1$, $\frac{5}{46368}a^{29}+\frac{514229}{46368}a^{5}+1$, $\frac{1}{46368}a^{25}+\frac{75025}{46368}a-1$, $\frac{5473}{23184}a^{29}-\frac{1}{15456}a^{28}-\frac{233}{21}a^{21}+\frac{10946}{21}a^{13}-\frac{257231}{23184}a^{5}-\frac{105937}{15456}a^{4}$, $\frac{2209}{2208}a^{31}-\frac{5473}{23184}a^{29}-\frac{799}{23184}a^{25}-47a^{23}+\frac{233}{21}a^{21}+\frac{34}{21}a^{17}+2208a^{15}-\frac{10946}{21}a^{13}-\frac{1597}{21}a^{9}-\frac{47}{2208}a^{7}+\frac{257231}{23184}a^{5}+\frac{17}{23184}a$, $\frac{13}{46368}a^{31}+\frac{10951}{46368}a^{29}+\frac{799}{23184}a^{26}-\frac{233}{21}a^{21}-\frac{34}{21}a^{18}+\frac{10946}{21}a^{13}+\frac{1597}{21}a^{10}+\frac{1346269}{46368}a^{7}-\frac{233}{46368}a^{5}-\frac{17}{23184}a^{2}$, $\frac{14335}{23184}a^{30}-\frac{1}{15456}a^{28}-\frac{610}{21}a^{22}+\frac{28657}{21}a^{14}-\frac{305}{23184}a^{6}-\frac{105937}{15456}a^{4}+1$, $\frac{1}{2208}a^{31}+\frac{47}{322}a^{27}+\frac{799}{23184}a^{25}-\frac{48}{7}a^{19}-\frac{34}{21}a^{17}+\frac{2255}{7}a^{11}+\frac{1597}{21}a^{9}+\frac{103729}{2208}a^{7}-\frac{1}{322}a^{3}-\frac{17}{23184}a$, $\frac{2209}{2208}a^{31}+\frac{2255}{15456}a^{28}+\frac{2585}{46368}a^{26}-47a^{23}-\frac{48}{7}a^{20}-\frac{55}{21}a^{18}+2208a^{15}+\frac{2255}{7}a^{12}+\frac{2584}{21}a^{10}-\frac{47}{2208}a^{7}-\frac{105985}{15456}a^{4}-\frac{55}{46368}a^{2}$, $\frac{1}{2208}a^{31}+\frac{2255}{15456}a^{28}-\frac{323}{5796}a^{26}-\frac{48}{7}a^{20}+\frac{55}{21}a^{18}+\frac{2255}{7}a^{12}-\frac{2584}{21}a^{10}+\frac{103729}{2208}a^{7}-\frac{105985}{15456}a^{4}+\frac{15181}{5796}a^{2}$, $\frac{28657}{46368}a^{30}-\frac{5}{46368}a^{29}+\frac{47}{322}a^{28}-\frac{323}{5796}a^{26}-\frac{1}{46368}a^{25}-\frac{799}{23184}a^{24}-\frac{610}{21}a^{22}-\frac{48}{7}a^{20}+\frac{55}{21}a^{18}+\frac{34}{21}a^{16}+\frac{28657}{21}a^{14}+\frac{2255}{7}a^{12}-\frac{2584}{21}a^{10}-\frac{1597}{21}a^{8}-\frac{1346879}{46368}a^{6}-\frac{514229}{46368}a^{5}-\frac{1}{322}a^{4}+\frac{15181}{5796}a^{2}-\frac{75025}{46368}a+\frac{17}{23184}$ 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:  \( 82239790500.5115 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 82239790500.5115 \cdot 20}{48\cdot\sqrt{520402924666472696020370152488960000000000000000}}\cr\approx \mathstrut & 0.280271400080522 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 47*x^24 + 2208*x^16 - 47*x^8 + 1)
 
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^32 - 47*x^24 + 2208*x^16 - 47*x^8 + 1, 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^32 - 47*x^24 + 2208*x^16 - 47*x^8 + 1);
 
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^32 - 47*x^24 + 2208*x^16 - 47*x^8 + 1);
 
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

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

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 32
The 32 conjugacy class representatives for $C_2^3\times C_4$
Character table for $C_2^3\times C_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(i, \sqrt{30})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-5}, \sqrt{-6})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{15})\), \(\Q(i, \sqrt{10})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{6}, \sqrt{-10})\), 4.0.18432.2, 4.4.18432.1, 4.0.51200.2, 4.4.51200.1, 4.4.460800.1, 4.0.460800.2, \(\Q(\zeta_{16})^+\), 4.0.2048.2, 8.0.3317760000.9, 8.0.3317760000.4, 8.0.3317760000.2, 8.0.40960000.1, 8.0.12960000.1, \(\Q(\zeta_{24})\), 8.0.3317760000.5, 8.0.3317760000.3, 8.0.3317760000.1, 8.8.3317760000.1, 8.0.207360000.2, 8.0.3317760000.6, 8.0.3317760000.8, 8.0.207360000.1, 8.0.3317760000.7, 8.0.1358954496.4, 8.0.10485760000.3, 8.0.849346560000.6, \(\Q(\zeta_{16})\), 8.0.849346560000.1, 8.8.849346560000.1, 8.8.849346560000.2, 8.0.849346560000.2, 8.0.212336640000.2, 8.0.212336640000.3, 8.0.212336640000.1, 8.0.212336640000.4, 8.0.849346560000.3, 8.0.849346560000.5, 8.0.10485760000.2, 8.0.10485760000.1, 8.0.212336640000.5, 8.8.212336640000.1, 8.0.2621440000.1, 8.8.2621440000.1, 8.0.339738624.2, 8.0.339738624.1, 8.0.212336640000.6, 8.0.212336640000.7, 8.0.1358954496.3, \(\Q(\zeta_{48})^+\), 8.0.849346560000.4, 8.8.849346560000.3, 16.0.11007531417600000000.1, 16.0.721389578983833600000000.3, 16.0.721389578983833600000000.7, 16.0.721389578983833600000000.1, 16.0.109951162777600000000.1, \(\Q(\zeta_{48})\), 16.0.721389578983833600000000.8, 16.0.721389578983833600000000.9, 16.0.721389578983833600000000.5, 16.0.721389578983833600000000.4, 16.16.721389578983833600000000.1, 16.0.721389578983833600000000.6, 16.0.721389578983833600000000.2, 16.0.45086848686489600000000.1, 16.0.45086848686489600000000.2

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 R R R ${\href{/padicField/7.2.0.1}{2} }^{16}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{16}$ ${\href{/padicField/19.4.0.1}{4} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{16}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{16}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{8}$

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

# 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
\(2\) Copy content Toggle raw display 2.16.48.1$x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516$$8$$2$$48$$C_4\times C_2^2$$[2, 3, 4]^{2}$
2.16.48.1$x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516$$8$$2$$48$$C_4\times C_2^2$$[2, 3, 4]^{2}$
\(3\) 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}$
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 5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$