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Properties

Label 16.0.260...625.4
Degree $16$
Signature $[0, 8]$
Discriminant $2.609\times 10^{25}$
Root discriminant \(38.77\)
Ramified primes $3,5,7,41$
Class number $168$
Class group [2, 84]
Galois group $C_2 \times (C_2^2:C_4)$ (as 16T21)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 10*x^14 + 129*x^12 + 128*x^11 - 262*x^10 - 1405*x^9 - 604*x^8 + 2918*x^7 + 16953*x^6 + 35934*x^5 + 72154*x^4 + 89871*x^3 + 113604*x^2 + 78021*x + 62631)
 
gp: K = bnfinit(y^16 - 2*y^15 - 10*y^14 + 129*y^12 + 128*y^11 - 262*y^10 - 1405*y^9 - 604*y^8 + 2918*y^7 + 16953*y^6 + 35934*y^5 + 72154*y^4 + 89871*y^3 + 113604*y^2 + 78021*y + 62631, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 10*x^14 + 129*x^12 + 128*x^11 - 262*x^10 - 1405*x^9 - 604*x^8 + 2918*x^7 + 16953*x^6 + 35934*x^5 + 72154*x^4 + 89871*x^3 + 113604*x^2 + 78021*x + 62631);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 10*x^14 + 129*x^12 + 128*x^11 - 262*x^10 - 1405*x^9 - 604*x^8 + 2918*x^7 + 16953*x^6 + 35934*x^5 + 72154*x^4 + 89871*x^3 + 113604*x^2 + 78021*x + 62631)
 

\( x^{16} - 2 x^{15} - 10 x^{14} + 129 x^{12} + 128 x^{11} - 262 x^{10} - 1405 x^{9} - 604 x^{8} + \cdots + 62631 \) 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:   \(26093349826855156494140625\) \(\medspace = 3^{8}\cdot 5^{12}\cdot 7^{8}\cdot 41^{4}\) 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:  \(38.77\)
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:  $3^{1/2}5^{3/4}7^{1/2}41^{1/2}\approx 98.11357012736104$
Ramified primes:   \(3\), \(5\), \(7\), \(41\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6}a^{12}-\frac{1}{3}a^{10}-\frac{1}{6}a^{9}-\frac{1}{6}a^{6}-\frac{1}{6}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{13}-\frac{1}{3}a^{11}-\frac{1}{6}a^{10}-\frac{1}{6}a^{7}-\frac{1}{6}a^{4}-\frac{1}{3}a^{3}-\frac{1}{2}a$, $\frac{1}{6}a^{14}-\frac{1}{6}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{6}a^{8}-\frac{1}{3}a^{6}-\frac{1}{6}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{6}a^{2}$, $\frac{1}{25\!\cdots\!86}a^{15}+\frac{21\!\cdots\!23}{43\!\cdots\!31}a^{14}+\frac{30\!\cdots\!65}{43\!\cdots\!31}a^{13}-\frac{18\!\cdots\!51}{12\!\cdots\!93}a^{12}+\frac{55\!\cdots\!48}{12\!\cdots\!93}a^{11}-\frac{12\!\cdots\!32}{43\!\cdots\!31}a^{10}-\frac{12\!\cdots\!13}{43\!\cdots\!31}a^{9}-\frac{37\!\cdots\!69}{43\!\cdots\!31}a^{8}+\frac{23\!\cdots\!73}{12\!\cdots\!93}a^{7}+\frac{71\!\cdots\!34}{43\!\cdots\!31}a^{6}+\frac{13\!\cdots\!69}{12\!\cdots\!93}a^{5}-\frac{35\!\cdots\!41}{12\!\cdots\!93}a^{4}+\frac{12\!\cdots\!91}{43\!\cdots\!31}a^{3}+\frac{60\!\cdots\!95}{12\!\cdots\!93}a^{2}-\frac{17\!\cdots\!94}{43\!\cdots\!31}a+\frac{40\!\cdots\!85}{86\!\cdots\!62}$ 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:  $2$, $3$

Class group and class number

$C_{2}\times C_{84}$, which has order $168$

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:   \( -1 \)  (order $2$) 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:   $\frac{131271386802}{18\!\cdots\!41}a^{15}-\frac{307937933701}{18\!\cdots\!41}a^{14}-\frac{4405599924059}{56\!\cdots\!23}a^{13}+\frac{3304382968459}{56\!\cdots\!23}a^{12}+\frac{55418673486403}{56\!\cdots\!23}a^{11}+\frac{11511708662820}{18\!\cdots\!41}a^{10}-\frac{200424899694490}{56\!\cdots\!23}a^{9}-\frac{185838907877968}{18\!\cdots\!41}a^{8}+\frac{68225878182659}{56\!\cdots\!23}a^{7}+\frac{21\!\cdots\!12}{56\!\cdots\!23}a^{6}+\frac{20\!\cdots\!74}{18\!\cdots\!41}a^{5}+\frac{10\!\cdots\!16}{56\!\cdots\!23}a^{4}+\frac{57\!\cdots\!35}{18\!\cdots\!41}a^{3}+\frac{19\!\cdots\!78}{56\!\cdots\!23}a^{2}+\frac{52\!\cdots\!30}{18\!\cdots\!41}a+\frac{30\!\cdots\!90}{18\!\cdots\!41}$, $\frac{12\!\cdots\!24}{21\!\cdots\!33}a^{15}+\frac{11\!\cdots\!14}{73\!\cdots\!11}a^{14}-\frac{28\!\cdots\!05}{43\!\cdots\!66}a^{13}-\frac{21\!\cdots\!47}{14\!\cdots\!22}a^{12}+\frac{35\!\cdots\!57}{73\!\cdots\!11}a^{11}+\frac{93\!\cdots\!99}{43\!\cdots\!66}a^{10}-\frac{94\!\cdots\!13}{43\!\cdots\!66}a^{9}-\frac{70\!\cdots\!20}{73\!\cdots\!11}a^{8}-\frac{96\!\cdots\!05}{14\!\cdots\!22}a^{7}+\frac{13\!\cdots\!61}{43\!\cdots\!66}a^{6}+\frac{13\!\cdots\!67}{21\!\cdots\!33}a^{5}+\frac{20\!\cdots\!39}{14\!\cdots\!22}a^{4}+\frac{81\!\cdots\!29}{43\!\cdots\!66}a^{3}+\frac{64\!\cdots\!67}{21\!\cdots\!33}a^{2}+\frac{24\!\cdots\!81}{14\!\cdots\!22}a+\frac{43\!\cdots\!59}{14\!\cdots\!22}$, $\frac{13\!\cdots\!73}{14\!\cdots\!22}a^{15}+\frac{20\!\cdots\!98}{21\!\cdots\!33}a^{14}-\frac{10\!\cdots\!01}{43\!\cdots\!66}a^{13}-\frac{32\!\cdots\!27}{14\!\cdots\!22}a^{12}-\frac{36\!\cdots\!56}{73\!\cdots\!11}a^{11}+\frac{35\!\cdots\!11}{43\!\cdots\!66}a^{10}-\frac{36\!\cdots\!71}{43\!\cdots\!66}a^{9}-\frac{26\!\cdots\!28}{21\!\cdots\!33}a^{8}-\frac{23\!\cdots\!51}{43\!\cdots\!66}a^{7}+\frac{46\!\cdots\!31}{43\!\cdots\!66}a^{6}-\frac{73\!\cdots\!63}{21\!\cdots\!33}a^{5}+\frac{88\!\cdots\!71}{14\!\cdots\!22}a^{4}+\frac{11\!\cdots\!39}{43\!\cdots\!66}a^{3}+\frac{31\!\cdots\!14}{21\!\cdots\!33}a^{2}+\frac{46\!\cdots\!63}{14\!\cdots\!22}a+\frac{13\!\cdots\!00}{73\!\cdots\!11}$, $\frac{34\!\cdots\!87}{86\!\cdots\!62}a^{15}-\frac{25\!\cdots\!41}{12\!\cdots\!93}a^{14}-\frac{37\!\cdots\!11}{43\!\cdots\!31}a^{13}+\frac{11\!\cdots\!13}{12\!\cdots\!93}a^{12}+\frac{61\!\cdots\!06}{12\!\cdots\!93}a^{11}-\frac{11\!\cdots\!21}{12\!\cdots\!93}a^{10}-\frac{21\!\cdots\!80}{12\!\cdots\!93}a^{9}-\frac{40\!\cdots\!18}{12\!\cdots\!93}a^{8}+\frac{50\!\cdots\!31}{43\!\cdots\!31}a^{7}+\frac{17\!\cdots\!59}{12\!\cdots\!93}a^{6}+\frac{56\!\cdots\!78}{12\!\cdots\!93}a^{5}-\frac{63\!\cdots\!58}{12\!\cdots\!93}a^{4}-\frac{60\!\cdots\!45}{12\!\cdots\!93}a^{3}-\frac{33\!\cdots\!09}{12\!\cdots\!93}a^{2}-\frac{54\!\cdots\!54}{43\!\cdots\!31}a-\frac{48\!\cdots\!15}{86\!\cdots\!62}$, $\frac{26\!\cdots\!65}{25\!\cdots\!86}a^{15}-\frac{15\!\cdots\!62}{43\!\cdots\!31}a^{14}-\frac{32\!\cdots\!57}{43\!\cdots\!31}a^{13}+\frac{34\!\cdots\!07}{12\!\cdots\!93}a^{12}+\frac{12\!\cdots\!53}{12\!\cdots\!93}a^{11}-\frac{41\!\cdots\!70}{43\!\cdots\!31}a^{10}-\frac{18\!\cdots\!59}{43\!\cdots\!31}a^{9}-\frac{18\!\cdots\!07}{43\!\cdots\!31}a^{8}+\frac{11\!\cdots\!99}{12\!\cdots\!93}a^{7}+\frac{16\!\cdots\!72}{43\!\cdots\!31}a^{6}+\frac{11\!\cdots\!30}{12\!\cdots\!93}a^{5}+\frac{17\!\cdots\!82}{12\!\cdots\!93}a^{4}+\frac{60\!\cdots\!79}{43\!\cdots\!31}a^{3}+\frac{17\!\cdots\!15}{12\!\cdots\!93}a^{2}+\frac{36\!\cdots\!15}{43\!\cdots\!31}a+\frac{40\!\cdots\!93}{86\!\cdots\!62}$, $\frac{44\!\cdots\!97}{43\!\cdots\!31}a^{15}-\frac{47\!\cdots\!52}{43\!\cdots\!31}a^{14}+\frac{15\!\cdots\!22}{12\!\cdots\!93}a^{13}+\frac{30\!\cdots\!90}{43\!\cdots\!31}a^{12}+\frac{14\!\cdots\!98}{12\!\cdots\!93}a^{11}-\frac{12\!\cdots\!06}{12\!\cdots\!93}a^{10}-\frac{37\!\cdots\!80}{43\!\cdots\!31}a^{9}+\frac{20\!\cdots\!30}{43\!\cdots\!31}a^{8}+\frac{13\!\cdots\!62}{12\!\cdots\!93}a^{7}+\frac{29\!\cdots\!78}{43\!\cdots\!31}a^{6}-\frac{72\!\cdots\!46}{43\!\cdots\!31}a^{5}-\frac{14\!\cdots\!71}{12\!\cdots\!93}a^{4}-\frac{26\!\cdots\!72}{12\!\cdots\!93}a^{3}-\frac{16\!\cdots\!45}{43\!\cdots\!31}a^{2}-\frac{12\!\cdots\!92}{43\!\cdots\!31}a-\frac{84\!\cdots\!05}{43\!\cdots\!31}$, $\frac{52\!\cdots\!69}{86\!\cdots\!62}a^{15}-\frac{22\!\cdots\!54}{12\!\cdots\!93}a^{14}-\frac{62\!\cdots\!69}{12\!\cdots\!93}a^{13}+\frac{11\!\cdots\!48}{12\!\cdots\!93}a^{12}+\frac{29\!\cdots\!63}{43\!\cdots\!31}a^{11}-\frac{18\!\cdots\!27}{12\!\cdots\!93}a^{10}-\frac{10\!\cdots\!91}{43\!\cdots\!31}a^{9}-\frac{60\!\cdots\!99}{12\!\cdots\!93}a^{8}+\frac{80\!\cdots\!23}{12\!\cdots\!93}a^{7}+\frac{10\!\cdots\!83}{43\!\cdots\!31}a^{6}+\frac{82\!\cdots\!66}{12\!\cdots\!93}a^{5}+\frac{28\!\cdots\!91}{43\!\cdots\!31}a^{4}+\frac{12\!\cdots\!77}{12\!\cdots\!93}a^{3}+\frac{16\!\cdots\!25}{43\!\cdots\!31}a^{2}+\frac{21\!\cdots\!13}{43\!\cdots\!31}a-\frac{13\!\cdots\!63}{86\!\cdots\!62}$ Copy content Toggle raw display
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:  \( 10253.363436451587 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10253.363436451587 \cdot 168}{2\cdot\sqrt{26093349826855156494140625}}\cr\approx \mathstrut & 0.409562010276402 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 10*x^14 + 129*x^12 + 128*x^11 - 262*x^10 - 1405*x^9 - 604*x^8 + 2918*x^7 + 16953*x^6 + 35934*x^5 + 72154*x^4 + 89871*x^3 + 113604*x^2 + 78021*x + 62631)
 
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 - 2*x^15 - 10*x^14 + 129*x^12 + 128*x^11 - 262*x^10 - 1405*x^9 - 604*x^8 + 2918*x^7 + 16953*x^6 + 35934*x^5 + 72154*x^4 + 89871*x^3 + 113604*x^2 + 78021*x + 62631, 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 - 2*x^15 - 10*x^14 + 129*x^12 + 128*x^11 - 262*x^10 - 1405*x^9 - 604*x^8 + 2918*x^7 + 16953*x^6 + 35934*x^5 + 72154*x^4 + 89871*x^3 + 113604*x^2 + 78021*x + 62631);
 
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 - 2*x^15 - 10*x^14 + 129*x^12 + 128*x^11 - 262*x^10 - 1405*x^9 - 604*x^8 + 2918*x^7 + 16953*x^6 + 35934*x^5 + 72154*x^4 + 89871*x^3 + 113604*x^2 + 78021*x + 62631);
 
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

$C_2^3:C_4$ (as 16T21):

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 32
The 20 conjugacy class representatives for $C_2 \times (C_2^2:C_4)$
Character table for $C_2 \times (C_2^2:C_4)$

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), 4.0.55125.1, \(\Q(\zeta_{15})^+\), 4.0.452025.1, 4.4.9225.1, \(\Q(\sqrt{5}, \sqrt{-7})\), 4.4.5125.1, 4.0.251125.1, 8.0.63063765625.1, 8.0.3038765625.2, 8.0.204326600625.2, 8.0.5108165015625.9, 8.8.2127515625.1, 8.0.5108165015625.15, 8.0.5108165015625.14

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

Galois closure: data not computed
Degree 16 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 ${\href{/padicField/2.4.0.1}{4} }^{4}$ R R R ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ R ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
\(5\) 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}$
\(7\) 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}$
\(41\) Copy content Toggle raw display 41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$