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Properties

Label 32.0.361...000.2
Degree $32$
Signature $[0, 16]$
Discriminant $3.613\times 10^{50}$
Root discriminant \(38.01\)
Ramified primes $2,3,5,29$
Class number not computed
Class group not computed
Galois group $C_2^4:C_4$ (as 32T262)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^30 - 56*x^28 + 824*x^26 + 992*x^24 - 23552*x^22 + 121088*x^20 - 560896*x^18 + 1820416*x^16 - 2243584*x^14 + 1937408*x^12 - 1507328*x^10 + 253952*x^8 + 843776*x^6 - 229376*x^4 - 32768*x^2 + 65536)
 
gp: K = bnfinit(y^32 - 2*y^30 - 56*y^28 + 824*y^26 + 992*y^24 - 23552*y^22 + 121088*y^20 - 560896*y^18 + 1820416*y^16 - 2243584*y^14 + 1937408*y^12 - 1507328*y^10 + 253952*y^8 + 843776*y^6 - 229376*y^4 - 32768*y^2 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 2*x^30 - 56*x^28 + 824*x^26 + 992*x^24 - 23552*x^22 + 121088*x^20 - 560896*x^18 + 1820416*x^16 - 2243584*x^14 + 1937408*x^12 - 1507328*x^10 + 253952*x^8 + 843776*x^6 - 229376*x^4 - 32768*x^2 + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 2*x^30 - 56*x^28 + 824*x^26 + 992*x^24 - 23552*x^22 + 121088*x^20 - 560896*x^18 + 1820416*x^16 - 2243584*x^14 + 1937408*x^12 - 1507328*x^10 + 253952*x^8 + 843776*x^6 - 229376*x^4 - 32768*x^2 + 65536)
 

\( x^{32} - 2 x^{30} - 56 x^{28} + 824 x^{26} + 992 x^{24} - 23552 x^{22} + 121088 x^{20} - 560896 x^{18} + \cdots + 65536 \) 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:   \(361280028614042283247927296000000000000000000000000\) \(\medspace = 2^{48}\cdot 3^{16}\cdot 5^{24}\cdot 29^{8}\) 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:  \(38.01\)
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:  $2^{3/2}3^{1/2}5^{3/4}29^{1/2}\approx 88.21290473450735$
Ramified primes:   \(2\), \(3\), \(5\), \(29\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{768}a^{16}+\frac{1}{384}a^{14}-\frac{1}{96}a^{10}-\frac{1}{48}a^{8}-\frac{1}{24}a^{6}+\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{768}a^{17}+\frac{1}{384}a^{15}-\frac{1}{96}a^{11}-\frac{1}{48}a^{9}-\frac{1}{24}a^{7}+\frac{1}{6}a^{3}+\frac{1}{3}a$, $\frac{1}{1536}a^{18}-\frac{1}{384}a^{14}-\frac{1}{192}a^{12}+\frac{1}{24}a^{6}+\frac{1}{12}a^{4}-\frac{1}{3}$, $\frac{1}{1536}a^{19}-\frac{1}{384}a^{15}-\frac{1}{192}a^{13}+\frac{1}{24}a^{7}+\frac{1}{12}a^{5}-\frac{1}{3}a$, $\frac{1}{12288}a^{20}+\frac{5}{384}a^{10}+\frac{1}{12}$, $\frac{1}{12288}a^{21}+\frac{5}{384}a^{11}+\frac{1}{12}a$, $\frac{1}{24576}a^{22}+\frac{5}{768}a^{12}+\frac{1}{24}a^{2}$, $\frac{1}{24576}a^{23}+\frac{5}{768}a^{13}+\frac{1}{24}a^{3}$, $\frac{1}{5898240}a^{24}-\frac{1}{196608}a^{22}+\frac{1}{32768}a^{20}+\frac{41}{184320}a^{18}+\frac{1}{6144}a^{16}+\frac{11}{12288}a^{14}-\frac{419}{92160}a^{12}+\frac{35}{3072}a^{10}-\frac{7}{384}a^{8}+\frac{161}{2880}a^{6}+\frac{11}{384}a^{4}-\frac{11}{64}a^{2}-\frac{599}{1440}$, $\frac{1}{5898240}a^{25}-\frac{1}{196608}a^{23}+\frac{1}{32768}a^{21}+\frac{41}{184320}a^{19}+\frac{1}{6144}a^{17}+\frac{11}{12288}a^{15}-\frac{419}{92160}a^{13}+\frac{35}{3072}a^{11}-\frac{7}{384}a^{9}+\frac{161}{2880}a^{7}+\frac{11}{384}a^{5}-\frac{11}{64}a^{3}-\frac{599}{1440}a$, $\frac{1}{342097920}a^{26}+\frac{1}{28508160}a^{24}+\frac{35}{2850816}a^{22}+\frac{389}{42762240}a^{20}+\frac{21}{148480}a^{18}+\frac{53}{237568}a^{16}-\frac{3517}{2672640}a^{14}+\frac{761}{445440}a^{12}-\frac{653}{89088}a^{10}-\frac{391}{41760}a^{8}+\frac{881}{37120}a^{6}+\frac{35}{5568}a^{4}-\frac{5797}{41760}a^{2}-\frac{2513}{13920}$, $\frac{1}{342097920}a^{27}+\frac{1}{28508160}a^{25}+\frac{35}{2850816}a^{23}+\frac{389}{42762240}a^{21}+\frac{21}{148480}a^{19}+\frac{53}{237568}a^{17}-\frac{3517}{2672640}a^{15}+\frac{761}{445440}a^{13}-\frac{653}{89088}a^{11}-\frac{391}{41760}a^{9}+\frac{881}{37120}a^{7}+\frac{35}{5568}a^{5}-\frac{5797}{41760}a^{3}-\frac{2513}{13920}a$, $\frac{1}{3067934146560}a^{28}-\frac{61}{51132235776}a^{26}+\frac{21589}{255661178880}a^{24}+\frac{2103697}{191745884160}a^{22}+\frac{29465}{4261019648}a^{20}-\frac{4520329}{31957647360}a^{18}+\frac{13057283}{23968235520}a^{16}-\frac{1966453}{532627456}a^{14}+\frac{1974939}{665784320}a^{12}-\frac{79457851}{5992058880}a^{10}+\frac{1500217}{66578432}a^{8}+\frac{10249163}{249669120}a^{6}+\frac{70804351}{749007360}a^{4}+\frac{233107}{12483456}a^{2}+\frac{6014539}{62417280}$, $\frac{1}{3067934146560}a^{29}-\frac{61}{51132235776}a^{27}+\frac{21589}{255661178880}a^{25}+\frac{2103697}{191745884160}a^{23}+\frac{29465}{4261019648}a^{21}-\frac{4520329}{31957647360}a^{19}+\frac{13057283}{23968235520}a^{17}-\frac{1966453}{532627456}a^{15}+\frac{1974939}{665784320}a^{13}-\frac{79457851}{5992058880}a^{11}+\frac{1500217}{66578432}a^{9}+\frac{10249163}{249669120}a^{7}+\frac{70804351}{749007360}a^{5}+\frac{233107}{12483456}a^{3}+\frac{6014539}{62417280}a$, $\frac{1}{951059585433600}a^{30}-\frac{1}{7669835366400}a^{28}+\frac{10907}{26418321817600}a^{26}-\frac{289063}{11888244817920}a^{24}+\frac{834797477}{59441224089600}a^{22}-\frac{3838369}{341616230400}a^{20}+\frac{78744139}{1486030602240}a^{18}+\frac{3148591843}{7430153011200}a^{16}-\frac{131615707}{619179417600}a^{14}-\frac{2496327887}{371507650560}a^{12}-\frac{4970558263}{928769126400}a^{10}+\frac{9887309}{25799142400}a^{8}-\frac{996093481}{46438456320}a^{6}-\frac{4077201151}{58048070400}a^{4}+\frac{31126021}{624172800}a^{2}+\frac{1088448923}{2418669600}$, $\frac{1}{951059585433600}a^{31}-\frac{1}{7669835366400}a^{29}+\frac{10907}{26418321817600}a^{27}-\frac{289063}{11888244817920}a^{25}+\frac{834797477}{59441224089600}a^{23}-\frac{3838369}{341616230400}a^{21}+\frac{78744139}{1486030602240}a^{19}+\frac{3148591843}{7430153011200}a^{17}-\frac{131615707}{619179417600}a^{15}-\frac{2496327887}{371507650560}a^{13}-\frac{4970558263}{928769126400}a^{11}+\frac{9887309}{25799142400}a^{9}-\frac{996093481}{46438456320}a^{7}-\frac{4077201151}{58048070400}a^{5}+\frac{31126021}{624172800}a^{3}+\frac{1088448923}{2418669600}a$ 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:  $2$

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:   \( -\frac{172379353}{158509930905600} a^{30} + \frac{13896151}{15339670732800} a^{28} + \frac{467216543}{7430153011200} a^{26} - \frac{3917610745}{4755297927168} a^{24} - \frac{31110261769}{14860306022400} a^{22} + \frac{710042366819}{29720612044800} a^{20} - \frac{303994918351}{2972061204480} a^{18} + \frac{872091803479}{1857538252800} a^{16} - \frac{4967698026809}{3715076505600} a^{14} + \frac{10460556253}{23219228160} a^{12} - \frac{15388165999}{48882585600} a^{10} + \frac{266539075949}{464384563200} a^{8} + \frac{2147311417}{5804807040} a^{6} - \frac{260234611}{1967731200} a^{4} - \frac{53512049}{24638400} a^{2} + \frac{1230590717}{29024035200} \)  (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:  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 - 2*x^30 - 56*x^28 + 824*x^26 + 992*x^24 - 23552*x^22 + 121088*x^20 - 560896*x^18 + 1820416*x^16 - 2243584*x^14 + 1937408*x^12 - 1507328*x^10 + 253952*x^8 + 843776*x^6 - 229376*x^4 - 32768*x^2 + 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^32 - 2*x^30 - 56*x^28 + 824*x^26 + 992*x^24 - 23552*x^22 + 121088*x^20 - 560896*x^18 + 1820416*x^16 - 2243584*x^14 + 1937408*x^12 - 1507328*x^10 + 253952*x^8 + 843776*x^6 - 229376*x^4 - 32768*x^2 + 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^32 - 2*x^30 - 56*x^28 + 824*x^26 + 992*x^24 - 23552*x^22 + 121088*x^20 - 560896*x^18 + 1820416*x^16 - 2243584*x^14 + 1937408*x^12 - 1507328*x^10 + 253952*x^8 + 843776*x^6 - 229376*x^4 - 32768*x^2 + 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^32 - 2*x^30 - 56*x^28 + 824*x^26 + 992*x^24 - 23552*x^22 + 121088*x^20 - 560896*x^18 + 1820416*x^16 - 2243584*x^14 + 1937408*x^12 - 1507328*x^10 + 253952*x^8 + 843776*x^6 - 229376*x^4 - 32768*x^2 + 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

$C_2^4:C_4$ (as 32T262):

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)
 
A solvable group of order 64
The 40 conjugacy class representatives for $C_2^4:C_4$
Character table for $C_2^4:C_4$

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-15}) \), 4.4.32625.1, 4.0.232000.1, 4.4.2088000.3, 4.0.3625.1, 4.0.8000.2, \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), 4.4.72000.1, \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.417600.1, 4.4.46400.1, 4.0.6525.1, 4.4.725.1, \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.207360000.1, 8.0.174389760000.27, 8.0.174389760000.12, 8.0.174389760000.25, 8.8.2152960000.1, 8.0.174389760000.40, 8.0.42575625.1, 8.0.4359744000000.86, 8.0.4359744000000.88, 8.0.5184000000.1, 8.0.5184000000.5, 8.8.4359744000000.2, 8.0.53824000000.4, 8.0.64000000.2, 8.8.5184000000.1, 8.0.4359744000000.21, 8.0.4359744000000.45, 8.0.4359744000000.70, 8.0.4359744000000.49, 8.0.53824000000.1, 8.8.4359744000000.4, 8.0.53824000000.12, 8.8.4359744000000.9, 8.0.4359744000000.58, 8.0.1064390625.1, 8.0.1064390625.3, 8.0.4359744000000.40, 8.0.53824000000.6, 8.8.1064390625.1, 8.0.13140625.1, 8.8.4359744000000.5, 8.0.4359744000000.95, 8.0.1064390625.2, \(\Q(\zeta_{15})\), 8.0.5184000000.3, 16.0.30411788392857600000000.2, 16.0.19007367745536000000000000.5, 16.0.26873856000000000000.2, 16.0.19007367745536000000000000.8, 16.0.19007367745536000000000000.17, 16.0.19007367745536000000000000.15, 16.0.19007367745536000000000000.13, 16.0.19007367745536000000000000.7, 16.0.19007367745536000000000000.6, 16.0.2897022976000000000000.1, 16.16.19007367745536000000000000.2, 16.0.19007367745536000000000000.19, 16.0.19007367745536000000000000.10, 16.0.19007367745536000000000000.14, 16.0.1132927402587890625.1

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]
 

Sibling fields

Degree 32 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.2.0.1}{2} }^{16}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ R ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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

# 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 2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
\(3\) 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 5.8.6.1$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}$
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}$
\(29\) Copy content Toggle raw display 29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$