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Properties

Label 32.0.156...000.3
Degree $32$
Signature $[0, 16]$
Discriminant $1.569\times 10^{59}$
Root discriminant \(70.77\)
Ramified primes $2,5,7$
Class number not computed
Class group not computed
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 16*x^30 + 176*x^28 + 1664*x^26 + 14591*x^24 + 88112*x^22 + 455328*x^20 + 2102272*x^18 + 8473857*x^16 + 25593824*x^14 + 70660512*x^12 + 165613696*x^10 + 264513791*x^8 + 2076688*x^6 + 16304*x^4 + 128*x^2 + 1)
 
gp: K = bnfinit(y^32 + 16*y^30 + 176*y^28 + 1664*y^26 + 14591*y^24 + 88112*y^22 + 455328*y^20 + 2102272*y^18 + 8473857*y^16 + 25593824*y^14 + 70660512*y^12 + 165613696*y^10 + 264513791*y^8 + 2076688*y^6 + 16304*y^4 + 128*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 16*x^30 + 176*x^28 + 1664*x^26 + 14591*x^24 + 88112*x^22 + 455328*x^20 + 2102272*x^18 + 8473857*x^16 + 25593824*x^14 + 70660512*x^12 + 165613696*x^10 + 264513791*x^8 + 2076688*x^6 + 16304*x^4 + 128*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 16*x^30 + 176*x^28 + 1664*x^26 + 14591*x^24 + 88112*x^22 + 455328*x^20 + 2102272*x^18 + 8473857*x^16 + 25593824*x^14 + 70660512*x^12 + 165613696*x^10 + 264513791*x^8 + 2076688*x^6 + 16304*x^4 + 128*x^2 + 1)
 

\( x^{32} + 16 x^{30} + 176 x^{28} + 1664 x^{26} + 14591 x^{24} + 88112 x^{22} + 455328 x^{20} + 2102272 x^{18} + \cdots + 1 \) 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:   \(156938077449417789520626992646455296000000000000000000000000\) \(\medspace = 2^{96}\cdot 5^{24}\cdot 7^{16}\) 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:  \(70.77\)
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:  $2^{3}5^{3/4}7^{1/2}\approx 70.7728215461241$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(560=2^{4}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(391,·)$, $\chi_{560}(393,·)$, $\chi_{560}(139,·)$, $\chi_{560}(141,·)$, $\chi_{560}(531,·)$, $\chi_{560}(533,·)$, $\chi_{560}(279,·)$, $\chi_{560}(281,·)$, $\chi_{560}(27,·)$, $\chi_{560}(29,·)$, $\chi_{560}(419,·)$, $\chi_{560}(421,·)$, $\chi_{560}(167,·)$, $\chi_{560}(169,·)$, $\chi_{560}(559,·)$, $\chi_{560}(307,·)$, $\chi_{560}(309,·)$, $\chi_{560}(57,·)$, $\chi_{560}(447,·)$, $\chi_{560}(449,·)$, $\chi_{560}(197,·)$, $\chi_{560}(337,·)$, $\chi_{560}(83,·)$, $\chi_{560}(477,·)$, $\chi_{560}(223,·)$, $\chi_{560}(363,·)$, $\chi_{560}(111,·)$, $\chi_{560}(113,·)$, $\chi_{560}(503,·)$, $\chi_{560}(251,·)$, $\chi_{560}(253,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{14}-\frac{1}{3}a^{12}-\frac{1}{3}a^{8}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{15}-\frac{1}{3}a^{13}-\frac{1}{3}a^{9}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{93}a^{18}+\frac{1}{93}a^{16}-\frac{10}{31}a^{14}+\frac{1}{93}a^{12}+\frac{32}{93}a^{10}-\frac{5}{93}a^{8}+\frac{26}{93}a^{6}+\frac{26}{93}a^{4}-\frac{12}{31}a^{2}+\frac{26}{93}$, $\frac{1}{93}a^{19}+\frac{1}{93}a^{17}-\frac{10}{31}a^{15}+\frac{1}{93}a^{13}+\frac{32}{93}a^{11}-\frac{5}{93}a^{9}+\frac{26}{93}a^{7}+\frac{26}{93}a^{5}-\frac{12}{31}a^{3}+\frac{26}{93}a$, $\frac{1}{93}a^{20}-\frac{37}{93}a^{10}+\frac{5}{93}$, $\frac{1}{93}a^{21}-\frac{37}{93}a^{11}+\frac{5}{93}a$, $\frac{1}{93}a^{22}-\frac{37}{93}a^{12}+\frac{5}{93}a^{2}$, $\frac{1}{93}a^{23}-\frac{37}{93}a^{13}+\frac{5}{93}a^{3}$, $\frac{1}{93}a^{24}-\frac{37}{93}a^{14}+\frac{5}{93}a^{4}$, $\frac{1}{93}a^{25}-\frac{37}{93}a^{15}+\frac{5}{93}a^{5}$, $\frac{1}{66\!\cdots\!63}a^{26}-\frac{172721887392179}{66\!\cdots\!63}a^{24}+\frac{23017152462407}{66\!\cdots\!63}a^{22}-\frac{1839142177020}{717853282661791}a^{20}-\frac{125845884110966}{66\!\cdots\!63}a^{18}-\frac{52\!\cdots\!54}{66\!\cdots\!63}a^{16}-\frac{88\!\cdots\!66}{22\!\cdots\!21}a^{14}+\frac{87\!\cdots\!26}{22\!\cdots\!21}a^{12}-\frac{10\!\cdots\!86}{66\!\cdots\!63}a^{10}-\frac{71\!\cdots\!48}{66\!\cdots\!63}a^{8}-\frac{17\!\cdots\!02}{66\!\cdots\!63}a^{6}-\frac{29\!\cdots\!84}{66\!\cdots\!63}a^{4}-\frac{62\!\cdots\!42}{22\!\cdots\!21}a^{2}+\frac{30\!\cdots\!85}{66\!\cdots\!63}$, $\frac{1}{66\!\cdots\!63}a^{27}-\frac{172721887392179}{66\!\cdots\!63}a^{25}+\frac{23017152462407}{66\!\cdots\!63}a^{23}-\frac{1839142177020}{717853282661791}a^{21}-\frac{125845884110966}{66\!\cdots\!63}a^{19}-\frac{52\!\cdots\!54}{66\!\cdots\!63}a^{17}-\frac{88\!\cdots\!66}{22\!\cdots\!21}a^{15}+\frac{87\!\cdots\!26}{22\!\cdots\!21}a^{13}-\frac{10\!\cdots\!86}{66\!\cdots\!63}a^{11}-\frac{71\!\cdots\!48}{66\!\cdots\!63}a^{9}-\frac{17\!\cdots\!02}{66\!\cdots\!63}a^{7}-\frac{29\!\cdots\!84}{66\!\cdots\!63}a^{5}-\frac{62\!\cdots\!42}{22\!\cdots\!21}a^{3}+\frac{30\!\cdots\!85}{66\!\cdots\!63}a$, $\frac{1}{20\!\cdots\!53}a^{28}+\frac{1}{20\!\cdots\!53}a^{26}-\frac{65\!\cdots\!55}{20\!\cdots\!53}a^{24}-\frac{3233118337864}{34\!\cdots\!47}a^{22}-\frac{127011925929511}{20\!\cdots\!53}a^{20}+\frac{17\!\cdots\!84}{68\!\cdots\!51}a^{18}+\frac{30\!\cdots\!46}{20\!\cdots\!53}a^{16}+\frac{25\!\cdots\!10}{68\!\cdots\!51}a^{14}-\frac{85\!\cdots\!70}{20\!\cdots\!53}a^{12}-\frac{18\!\cdots\!22}{68\!\cdots\!51}a^{10}-\frac{96\!\cdots\!52}{20\!\cdots\!53}a^{8}-\frac{87\!\cdots\!77}{20\!\cdots\!53}a^{6}-\frac{47\!\cdots\!50}{20\!\cdots\!53}a^{4}-\frac{10\!\cdots\!81}{20\!\cdots\!53}a^{2}-\frac{10\!\cdots\!78}{20\!\cdots\!53}$, $\frac{1}{20\!\cdots\!53}a^{29}+\frac{1}{20\!\cdots\!53}a^{27}-\frac{65\!\cdots\!55}{20\!\cdots\!53}a^{25}-\frac{3233118337864}{34\!\cdots\!47}a^{23}-\frac{127011925929511}{20\!\cdots\!53}a^{21}+\frac{17\!\cdots\!84}{68\!\cdots\!51}a^{19}+\frac{30\!\cdots\!46}{20\!\cdots\!53}a^{17}+\frac{25\!\cdots\!10}{68\!\cdots\!51}a^{15}-\frac{85\!\cdots\!70}{20\!\cdots\!53}a^{13}-\frac{18\!\cdots\!22}{68\!\cdots\!51}a^{11}-\frac{96\!\cdots\!52}{20\!\cdots\!53}a^{9}-\frac{87\!\cdots\!77}{20\!\cdots\!53}a^{7}-\frac{47\!\cdots\!50}{20\!\cdots\!53}a^{5}-\frac{10\!\cdots\!81}{20\!\cdots\!53}a^{3}-\frac{10\!\cdots\!78}{20\!\cdots\!53}a$, $\frac{1}{20\!\cdots\!53}a^{30}-\frac{49\!\cdots\!62}{20\!\cdots\!53}a^{20}+\frac{31\!\cdots\!63}{68\!\cdots\!51}a^{10}-\frac{21\!\cdots\!52}{20\!\cdots\!53}$, $\frac{1}{20\!\cdots\!53}a^{31}-\frac{49\!\cdots\!62}{20\!\cdots\!53}a^{21}+\frac{31\!\cdots\!63}{68\!\cdots\!51}a^{11}-\frac{21\!\cdots\!52}{20\!\cdots\!53}a$ 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:  $31$

Class group and class number

not computed

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:   \( -\frac{5442706427752448}{689857004637981151} a^{30} - \frac{977986311236768}{7751202299303159} a^{28} - \frac{957235993172535040}{689857004637981151} a^{26} - \frac{9049179774441913856}{689857004637981151} a^{24} - \frac{79343774303775186944}{689857004637981151} a^{22} - \frac{478947322750503885920}{689857004637981151} a^{20} - \frac{2474470009298462551552}{689857004637981151} a^{18} - \frac{11422688253086898160351}{689857004637981151} a^{16} - \frac{46031324951385669545984}{689857004637981151} a^{14} - \frac{138939352302113658151840}{689857004637981151} a^{12} - \frac{383496144175713634388992}{689857004637981151} a^{10} - \frac{898382161939518956758016}{689857004637981151} a^{8} - \frac{1432628844683515600944096}{689857004637981151} a^{6} - \frac{55374137717097372768}{689857004637981151} a^{4} - \frac{434736175916726784}{689857004637981151} a^{2} - \frac{3401691517345280}{689857004637981151} \)  (order $10$) 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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + 16*x^30 + 176*x^28 + 1664*x^26 + 14591*x^24 + 88112*x^22 + 455328*x^20 + 2102272*x^18 + 8473857*x^16 + 25593824*x^14 + 70660512*x^12 + 165613696*x^10 + 264513791*x^8 + 2076688*x^6 + 16304*x^4 + 128*x^2 + 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 + 16*x^30 + 176*x^28 + 1664*x^26 + 14591*x^24 + 88112*x^22 + 455328*x^20 + 2102272*x^18 + 8473857*x^16 + 25593824*x^14 + 70660512*x^12 + 165613696*x^10 + 264513791*x^8 + 2076688*x^6 + 16304*x^4 + 128*x^2 + 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 + 16*x^30 + 176*x^28 + 1664*x^26 + 14591*x^24 + 88112*x^22 + 455328*x^20 + 2102272*x^18 + 8473857*x^16 + 25593824*x^14 + 70660512*x^12 + 165613696*x^10 + 264513791*x^8 + 2076688*x^6 + 16304*x^4 + 128*x^2 + 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 + 16*x^30 + 176*x^28 + 1664*x^26 + 14591*x^24 + 88112*x^22 + 455328*x^20 + 2102272*x^18 + 8473857*x^16 + 25593824*x^14 + 70660512*x^12 + 165613696*x^10 + 264513791*x^8 + 2076688*x^6 + 16304*x^4 + 128*x^2 + 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

$C_2\times C_4^2$ (as 32T36):

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 $C_2\times C_4^2$
Character table for $C_2\times C_4^2$

Intermediate fields

\(\Q(\sqrt{35}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}, \sqrt{35})\), \(\Q(\zeta_{16})^+\), 4.4.2508800.1, \(\Q(\sqrt{10}, \sqrt{14})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{7})\), 4.4.51200.1, 4.4.100352.1, \(\Q(\sqrt{7}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{14})\), 4.0.12544000.1, 4.0.256000.2, 4.0.12544000.2, 4.0.256000.4, 4.0.98000.1, \(\Q(\zeta_{5})\), 4.0.392000.2, 4.0.8000.2, 8.8.25176309760000.2, 8.8.98344960000.1, 8.8.25176309760000.5, 8.8.2621440000.1, 8.8.40282095616.1, 8.8.6294077440000.1, 8.8.25176309760000.1, 8.0.629407744000000.70, 8.0.629407744000000.21, 8.0.9604000000.3, 8.0.2458624000000.3, 8.0.157351936000000.83, 8.0.65536000000.1, 8.0.2458624000000.6, 8.0.64000000.2, 8.0.629407744000000.69, 8.0.629407744000000.44, 8.0.2458624000000.1, 8.0.153664000000.6, 16.16.633846573131471257600000000.1, 16.0.396154108207169536000000000000.15, 16.0.6044831973376000000000000.5, 16.0.396154108207169536000000000000.14, 16.0.4294967296000000000000.1, 16.0.24759631762948096000000000000.5, 16.0.396154108207169536000000000000.5

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 ${\href{/padicField/3.4.0.1}{4} }^{8}$ R R ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.1.0.1}{1} }^{32}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{8}$

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 Deg $32$$8$$4$$96$
\(5\) Copy content Toggle raw display 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
\(7\) Copy content Toggle raw display 7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$