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Properties

Label 18.6.110...000.1
Degree $18$
Signature $[6, 6]$
Discriminant $1.107\times 10^{20}$
Root discriminant \(12.99\)
Ramified primes $2,5,7$
Class number $1$
Class group trivial
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 5*x^16 + 10*x^15 - 19*x^14 + 7*x^13 - 18*x^12 + 34*x^11 + 21*x^10 - 70*x^9 + 22*x^8 + 14*x^7 + 11*x^6 - 7*x^5 - 20*x^4 + 13*x^3 - x^2 + 4*x - 1)
 
gp: K = bnfinit(y^18 - 5*y^17 + 5*y^16 + 10*y^15 - 19*y^14 + 7*y^13 - 18*y^12 + 34*y^11 + 21*y^10 - 70*y^9 + 22*y^8 + 14*y^7 + 11*y^6 - 7*y^5 - 20*y^4 + 13*y^3 - y^2 + 4*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 5*x^17 + 5*x^16 + 10*x^15 - 19*x^14 + 7*x^13 - 18*x^12 + 34*x^11 + 21*x^10 - 70*x^9 + 22*x^8 + 14*x^7 + 11*x^6 - 7*x^5 - 20*x^4 + 13*x^3 - x^2 + 4*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 5*x^17 + 5*x^16 + 10*x^15 - 19*x^14 + 7*x^13 - 18*x^12 + 34*x^11 + 21*x^10 - 70*x^9 + 22*x^8 + 14*x^7 + 11*x^6 - 7*x^5 - 20*x^4 + 13*x^3 - x^2 + 4*x - 1)
 

\( x^{18} - 5 x^{17} + 5 x^{16} + 10 x^{15} - 19 x^{14} + 7 x^{13} - 18 x^{12} + 34 x^{11} + 21 x^{10} + \cdots - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[6, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(110730297608000000000\) \(\medspace = 2^{12}\cdot 5^{9}\cdot 7^{12}\) 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:  \(12.99\)
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:  $2\cdot 5^{1/2}7^{2/3}\approx 16.364912636128995$
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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $6$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{61668538231}a^{17}+\frac{17625470048}{61668538231}a^{16}-\frac{22926492839}{61668538231}a^{15}-\frac{16185322443}{61668538231}a^{14}+\frac{22119042450}{61668538231}a^{13}+\frac{8294293274}{61668538231}a^{12}+\frac{10064486282}{61668538231}a^{11}+\frac{15931590811}{61668538231}a^{10}+\frac{2163650743}{61668538231}a^{9}-\frac{28778891729}{61668538231}a^{8}+\frac{6174380026}{61668538231}a^{7}-\frac{28525476316}{61668538231}a^{6}+\frac{14193776766}{61668538231}a^{5}-\frac{27541996467}{61668538231}a^{4}-\frac{5729903377}{61668538231}a^{3}+\frac{25794475256}{61668538231}a^{2}-\frac{10134542531}{61668538231}a+\frac{5843535590}{61668538231}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

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:  $11$
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:   $\frac{66333386}{868570961}a^{17}-\frac{365100676}{868570961}a^{16}+\frac{424782055}{868570961}a^{15}+\frac{736147223}{868570961}a^{14}-\frac{1438653166}{868570961}a^{13}+\frac{217555210}{868570961}a^{12}-\frac{1436873251}{868570961}a^{11}+\frac{3300398656}{868570961}a^{10}+\frac{1691614404}{868570961}a^{9}-\frac{5027777039}{868570961}a^{8}+\frac{89117094}{868570961}a^{7}+\frac{601678269}{868570961}a^{6}+\frac{2853428308}{868570961}a^{5}-\frac{243960638}{868570961}a^{4}-\frac{1102729541}{868570961}a^{3}-\frac{390313163}{868570961}a^{2}-\frac{197091379}{868570961}a+\frac{618787545}{868570961}$, $\frac{12406323785}{61668538231}a^{17}-\frac{78445147492}{61668538231}a^{16}+\frac{146790642704}{61668538231}a^{15}+\frac{36341879418}{61668538231}a^{14}-\frac{419519635254}{61668538231}a^{13}+\frac{440261663389}{61668538231}a^{12}-\frac{296092333498}{61668538231}a^{11}+\frac{623447330334}{61668538231}a^{10}-\frac{326951187338}{61668538231}a^{9}-\frac{1202447737694}{61668538231}a^{8}+\frac{1563855559074}{61668538231}a^{7}-\frac{83422204366}{61668538231}a^{6}-\frac{516007960274}{61668538231}a^{5}-\frac{132271637372}{61668538231}a^{4}+\frac{64021991156}{61668538231}a^{3}+\frac{373337388120}{61668538231}a^{2}-\frac{238167704413}{61668538231}a+\frac{8441030223}{61668538231}$, $\frac{14780290404}{61668538231}a^{17}-\frac{66999898817}{61668538231}a^{16}+\frac{42240891166}{61668538231}a^{15}+\frac{170667555932}{61668538231}a^{14}-\frac{205598109058}{61668538231}a^{13}-\frac{6102098936}{61668538231}a^{12}-\frac{242635557243}{61668538231}a^{11}+\frac{410161850969}{61668538231}a^{10}+\frac{464318368965}{61668538231}a^{9}-\frac{841982199940}{61668538231}a^{8}-\frac{70148382466}{61668538231}a^{7}+\frac{230431652179}{61668538231}a^{6}+\frac{325446898396}{61668538231}a^{5}-\frac{143184517136}{61668538231}a^{4}-\frac{287578631055}{61668538231}a^{3}+\frac{201942524318}{61668538231}a^{2}-\frac{74189178139}{61668538231}a+\frac{40347204817}{61668538231}$, $\frac{740709126}{61668538231}a^{17}-\frac{14126466159}{61668538231}a^{16}+\frac{53966245793}{61668538231}a^{15}-\frac{46023488259}{61668538231}a^{14}-\frac{83053956562}{61668538231}a^{13}+\frac{162619968092}{61668538231}a^{12}-\frac{154833786292}{61668538231}a^{11}+\frac{302775854306}{61668538231}a^{10}-\frac{323887853668}{61668538231}a^{9}-\frac{131534560128}{61668538231}a^{8}+\frac{483334042283}{61668538231}a^{7}-\frac{335809301200}{61668538231}a^{6}+\frac{194068015663}{61668538231}a^{5}-\frac{86321924081}{61668538231}a^{4}-\frac{54775230431}{61668538231}a^{3}+\frac{96053870198}{61668538231}a^{2}-\frac{33302290670}{61668538231}a+\frac{44210249457}{61668538231}$, $\frac{6240487172}{61668538231}a^{17}-\frac{30744188929}{61668538231}a^{16}+\frac{24979044753}{61668538231}a^{15}+\frac{77707347607}{61668538231}a^{14}-\frac{102299517473}{61668538231}a^{13}-\frac{34915842456}{61668538231}a^{12}-\frac{82023331144}{61668538231}a^{11}+\frac{270526615488}{61668538231}a^{10}+\frac{146824282986}{61668538231}a^{9}-\frac{407821807615}{61668538231}a^{8}-\frac{190417085065}{61668538231}a^{7}+\frac{324919728549}{61668538231}a^{6}+\frac{278589177914}{61668538231}a^{5}-\frac{192502790489}{61668538231}a^{4}-\frac{176517467011}{61668538231}a^{3}+\frac{7444342587}{61668538231}a^{2}+\frac{123519427610}{61668538231}a+\frac{4799878054}{61668538231}$, $\frac{7011462952}{61668538231}a^{17}-\frac{39650432351}{61668538231}a^{16}+\frac{58681575584}{61668538231}a^{15}+\frac{42877741323}{61668538231}a^{14}-\frac{176614609796}{61668538231}a^{13}+\frac{161966441339}{61668538231}a^{12}-\frac{194329153994}{61668538231}a^{11}+\frac{290985741997}{61668538231}a^{10}+\frac{38924386723}{61668538231}a^{9}-\frac{576139572073}{61668538231}a^{8}+\frac{553657408217}{61668538231}a^{7}-\frac{158828795966}{61668538231}a^{6}-\frac{36470363902}{61668538231}a^{5}+\frac{109176916512}{61668538231}a^{4}-\frac{169353572797}{61668538231}a^{3}+\frac{90852624251}{61668538231}a^{2}-\frac{32878389106}{61668538231}a+\frac{61236523529}{61668538231}$, $\frac{18602741141}{61668538231}a^{17}-\frac{77672435685}{61668538231}a^{16}+\frac{15995759737}{61668538231}a^{15}+\frac{267052719186}{61668538231}a^{14}-\frac{217571456434}{61668538231}a^{13}-\frac{139781096199}{61668538231}a^{12}-\frac{205797113448}{61668538231}a^{11}+\frac{291268646143}{61668538231}a^{10}+\frac{948581668025}{61668538231}a^{9}-\frac{1052824783232}{61668538231}a^{8}-\frac{476928632977}{61668538231}a^{7}+\frac{568913678735}{61668538231}a^{6}+\frac{244182504635}{61668538231}a^{5}+\frac{47027562096}{61668538231}a^{4}-\frac{400192468084}{61668538231}a^{3}+\frac{43290970506}{61668538231}a^{2}+\frac{75349484789}{61668538231}a+\frac{35523674898}{61668538231}$, $\frac{299009394}{61668538231}a^{17}-\frac{10501597581}{61668538231}a^{16}+\frac{44341701502}{61668538231}a^{15}-\frac{38908386767}{61668538231}a^{14}-\frac{70245993911}{61668538231}a^{13}+\frac{113395694151}{61668538231}a^{12}-\frac{65094002633}{61668538231}a^{11}+\frac{233229505819}{61668538231}a^{10}-\frac{341713156827}{61668538231}a^{9}-\frac{34883366253}{61668538231}a^{8}+\frac{268412692607}{61668538231}a^{7}+\frac{2887214456}{61668538231}a^{6}-\frac{58015162437}{61668538231}a^{5}-\frac{161444748474}{61668538231}a^{4}+\frac{188299090613}{61668538231}a^{3}+\frac{12802957753}{61668538231}a^{2}+\frac{34263829145}{61668538231}a-\frac{48643371682}{61668538231}$, $\frac{12406494438}{61668538231}a^{17}-\frac{60388724942}{61668538231}a^{16}+\frac{47410639939}{61668538231}a^{15}+\frac{158006918860}{61668538231}a^{14}-\frac{223119381990}{61668538231}a^{13}-\frac{23346032449}{61668538231}a^{12}-\frac{168436046471}{61668538231}a^{11}+\frac{431363395127}{61668538231}a^{10}+\frac{437023127616}{61668538231}a^{9}-\frac{939267635198}{61668538231}a^{8}-\frac{93364611051}{61668538231}a^{7}+\frac{381218687581}{61668538231}a^{6}+\frac{341093385940}{61668538231}a^{5}-\frac{127284906427}{61668538231}a^{4}-\frac{376516580906}{61668538231}a^{3}+\frac{130990168584}{61668538231}a^{2}+\frac{74243132394}{61668538231}a+\frac{45056875223}{61668538231}$, $\frac{18887217211}{61668538231}a^{17}-\frac{77660284872}{61668538231}a^{16}+\frac{13461323525}{61668538231}a^{15}+\frac{264417819943}{61668538231}a^{14}-\frac{208964026585}{61668538231}a^{13}-\frac{124538790893}{61668538231}a^{12}-\frac{215124556707}{61668538231}a^{11}+\frac{273655355406}{61668538231}a^{10}+\frac{912258482445}{61668538231}a^{9}-\frac{1053311789583}{61668538231}a^{8}-\frac{398947384001}{61668538231}a^{7}+\frac{544665223832}{61668538231}a^{6}+\frac{293408081020}{61668538231}a^{5}-\frac{24060277326}{61668538231}a^{4}-\frac{458580893714}{61668538231}a^{3}+\frac{118588067148}{61668538231}a^{2}+\frac{56445502426}{61668538231}a+\frac{78247664934}{61668538231}$, $\frac{4400975982}{61668538231}a^{17}-\frac{20967218299}{61668538231}a^{16}+\frac{18003785973}{61668538231}a^{15}+\frac{43928512528}{61668538231}a^{14}-\frac{70176438885}{61668538231}a^{13}+\frac{32465164528}{61668538231}a^{12}-\frac{116926920504}{61668538231}a^{11}+\frac{140570104132}{61668538231}a^{10}+\frac{162079043097}{61668538231}a^{9}-\frac{294603025504}{61668538231}a^{8}+\frac{108637166908}{61668538231}a^{7}-\frac{169080744090}{61668538231}a^{6}+\frac{172106351893}{61668538231}a^{5}+\frac{102446979771}{61668538231}a^{4}-\frac{141335777532}{61668538231}a^{3}+\frac{51203659152}{61668538231}a^{2}-\frac{133644374423}{61668538231}a+\frac{95998652383}{61668538231}$ 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:  \( 742.586512103 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 742.586512103 \cdot 1}{2\cdot\sqrt{110730297608000000000}}\cr\approx \mathstrut & 0.138945028307 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 5*x^16 + 10*x^15 - 19*x^14 + 7*x^13 - 18*x^12 + 34*x^11 + 21*x^10 - 70*x^9 + 22*x^8 + 14*x^7 + 11*x^6 - 7*x^5 - 20*x^4 + 13*x^3 - x^2 + 4*x - 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^18 - 5*x^17 + 5*x^16 + 10*x^15 - 19*x^14 + 7*x^13 - 18*x^12 + 34*x^11 + 21*x^10 - 70*x^9 + 22*x^8 + 14*x^7 + 11*x^6 - 7*x^5 - 20*x^4 + 13*x^3 - x^2 + 4*x - 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^18 - 5*x^17 + 5*x^16 + 10*x^15 - 19*x^14 + 7*x^13 - 18*x^12 + 34*x^11 + 21*x^10 - 70*x^9 + 22*x^8 + 14*x^7 + 11*x^6 - 7*x^5 - 20*x^4 + 13*x^3 - x^2 + 4*x - 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^18 - 5*x^17 + 5*x^16 + 10*x^15 - 19*x^14 + 7*x^13 - 18*x^12 + 34*x^11 + 21*x^10 - 70*x^9 + 22*x^8 + 14*x^7 + 11*x^6 - 7*x^5 - 20*x^4 + 13*x^3 - x^2 + 4*x - 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

$C_6\times S_3$ (as 18T6):

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 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.980.1, \(\Q(\zeta_{7})^+\), 6.2.4802000.1, 6.6.300125.1, 9.3.941192000.1

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

Galois closure: data not computed
Degree 12 sibling: 12.0.153664000000.1
Degree 18 sibling: 18.0.56693912375296000000.1
Minimal sibling: 12.0.153664000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{3}$ R R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{9}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

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 2.6.0.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.26$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
\(5\) Copy content Toggle raw display 5.6.3.1$x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(7\) Copy content Toggle raw display 7.18.12.1$x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$$3$$6$$12$$C_6 \times C_3$$[\ ]_{3}^{6}$