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Properties

Label 18.0.261...000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-2.615\times 10^{23}$
Root discriminant \(20.00\)
Ramified primes $2,3,5$
Class number $1$
Class group trivial
Galois group $S_3\times D_6$ (as 18T29)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 + 11*x^15 - 39*x^14 + 69*x^13 + 13*x^12 - 114*x^11 + 225*x^10 - 219*x^9 + 174*x^8 - 72*x^7 - 24*x^6 + 45*x^5 + 36*x^4 + 18*x^3 - 9*x^2 + 9)
 
gp: K = bnfinit(y^18 - 3*y^17 + 6*y^16 + 11*y^15 - 39*y^14 + 69*y^13 + 13*y^12 - 114*y^11 + 225*y^10 - 219*y^9 + 174*y^8 - 72*y^7 - 24*y^6 + 45*y^5 + 36*y^4 + 18*y^3 - 9*y^2 + 9, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 6*x^16 + 11*x^15 - 39*x^14 + 69*x^13 + 13*x^12 - 114*x^11 + 225*x^10 - 219*x^9 + 174*x^8 - 72*x^7 - 24*x^6 + 45*x^5 + 36*x^4 + 18*x^3 - 9*x^2 + 9);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 + 6*x^16 + 11*x^15 - 39*x^14 + 69*x^13 + 13*x^12 - 114*x^11 + 225*x^10 - 219*x^9 + 174*x^8 - 72*x^7 - 24*x^6 + 45*x^5 + 36*x^4 + 18*x^3 - 9*x^2 + 9)
 

\( x^{18} - 3 x^{17} + 6 x^{16} + 11 x^{15} - 39 x^{14} + 69 x^{13} + 13 x^{12} - 114 x^{11} + 225 x^{10} + \cdots + 9 \) 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:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-261508830075000000000000\) \(\medspace = -\,2^{12}\cdot 3^{21}\cdot 5^{14}\) 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:  \(20.00\)
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:  $2\cdot 3^{7/6}5^{5/6}\approx 27.55157706505336$
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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{6}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{7}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{8}$, $\frac{1}{18}a^{12}+\frac{1}{9}a^{11}+\frac{1}{18}a^{10}-\frac{1}{9}a^{9}-\frac{7}{18}a^{8}-\frac{4}{9}a^{7}-\frac{1}{3}a^{6}-\frac{1}{2}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{18}a^{13}-\frac{1}{6}a^{11}+\frac{1}{9}a^{10}-\frac{1}{6}a^{9}+\frac{1}{3}a^{8}-\frac{1}{9}a^{7}-\frac{1}{3}a^{6}-\frac{1}{2}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{6}a$, $\frac{1}{18}a^{14}+\frac{1}{9}a^{11}+\frac{7}{18}a^{8}+\frac{1}{3}a^{7}-\frac{1}{2}a^{6}-\frac{1}{3}a^{5}-\frac{1}{6}a^{4}-\frac{1}{6}a^{2}-\frac{1}{2}$, $\frac{1}{18}a^{15}+\frac{1}{9}a^{11}-\frac{1}{9}a^{10}-\frac{1}{18}a^{9}+\frac{4}{9}a^{8}+\frac{7}{18}a^{7}-\frac{1}{3}a^{6}-\frac{1}{6}a^{5}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{6}a$, $\frac{1}{2034}a^{16}-\frac{20}{1017}a^{15}+\frac{2}{1017}a^{14}+\frac{22}{1017}a^{13}-\frac{4}{339}a^{12}-\frac{10}{113}a^{11}-\frac{7}{678}a^{10}+\frac{89}{1017}a^{9}-\frac{1003}{2034}a^{8}+\frac{2}{9}a^{7}+\frac{35}{678}a^{6}-\frac{38}{339}a^{5}-\frac{137}{678}a^{4}-\frac{5}{113}a^{3}-\frac{35}{678}a^{2}-\frac{115}{339}a-\frac{13}{113}$, $\frac{1}{122593412754}a^{17}-\frac{4300138}{20432235459}a^{16}-\frac{533249329}{20432235459}a^{15}+\frac{118964708}{20432235459}a^{14}+\frac{1647197455}{61296706377}a^{13}-\frac{901590283}{40864470918}a^{12}-\frac{623550793}{40864470918}a^{11}-\frac{14617852745}{122593412754}a^{10}+\frac{785633729}{13621490306}a^{9}+\frac{22452845659}{122593412754}a^{8}-\frac{5862005749}{122593412754}a^{7}-\frac{1811589844}{6810745153}a^{6}+\frac{2665003871}{13621490306}a^{5}-\frac{16918532651}{40864470918}a^{4}+\frac{19961955157}{40864470918}a^{3}+\frac{8996143597}{20432235459}a^{2}+\frac{5609387774}{20432235459}a+\frac{559823187}{13621490306}$ 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:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{4684510}{180816243} a^{17} - \frac{29800429}{542448729} a^{16} + \frac{16464866}{180816243} a^{15} + \frac{75091958}{180816243} a^{14} - \frac{414823214}{542448729} a^{13} + \frac{60709639}{60272081} a^{12} + \frac{317586424}{180816243} a^{11} - \frac{1498146262}{542448729} a^{10} + \frac{245739486}{60272081} a^{9} - \frac{94895721}{60272081} a^{8} - \frac{42923736}{60272081} a^{7} + \frac{673933994}{180816243} a^{6} - \frac{301670934}{60272081} a^{5} + \frac{502956883}{180816243} a^{4} + \frac{21354564}{60272081} a^{3} + \frac{74525384}{60272081} a^{2} + \frac{37916864}{60272081} a + \frac{20351970}{60272081} \)  (order $6$) 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:   $\frac{594445703}{20432235459}a^{17}-\frac{2690748097}{61296706377}a^{16}+\frac{1744463143}{40864470918}a^{15}+\frac{71642139227}{122593412754}a^{14}-\frac{81679250945}{122593412754}a^{13}+\frac{17793054491}{61296706377}a^{12}+\frac{46754550197}{13621490306}a^{11}-\frac{19674564187}{6810745153}a^{10}+\frac{93117344774}{61296706377}a^{9}+\frac{449955084877}{122593412754}a^{8}-\frac{71636197001}{13621490306}a^{7}+\frac{233355322969}{40864470918}a^{6}-\frac{72434799587}{20432235459}a^{5}-\frac{54346436773}{40864470918}a^{4}+\frac{150131800673}{40864470918}a^{3}+\frac{50149740797}{40864470918}a^{2}+\frac{3993949191}{6810745153}a+\frac{1953912963}{13621490306}$, $\frac{1694278769}{20432235459}a^{17}-\frac{12263232673}{61296706377}a^{16}+\frac{7445322682}{20432235459}a^{15}+\frac{142219288615}{122593412754}a^{14}-\frac{35562001311}{13621490306}a^{13}+\frac{483932249297}{122593412754}a^{12}+\frac{52277609229}{13621490306}a^{11}-\frac{955328476177}{122593412754}a^{10}+\frac{1601658626513}{122593412754}a^{9}-\frac{562475160389}{61296706377}a^{8}+\frac{145434868849}{20432235459}a^{7}-\frac{51867581819}{40864470918}a^{6}-\frac{148103789927}{40864470918}a^{5}+\frac{29560288012}{20432235459}a^{4}+\frac{76057214603}{20432235459}a^{3}+\frac{163926032383}{40864470918}a^{2}-\frac{3392427165}{13621490306}a-\frac{3259322810}{6810745153}$, $\frac{1817093447}{40864470918}a^{17}-\frac{11132578081}{122593412754}a^{16}+\frac{20373328405}{122593412754}a^{15}+\frac{41897085641}{61296706377}a^{14}-\frac{71548660783}{61296706377}a^{13}+\frac{74828842267}{40864470918}a^{12}+\frac{329386586917}{122593412754}a^{11}-\frac{23995539653}{6810745153}a^{10}+\frac{426462703531}{61296706377}a^{9}-\frac{57854891114}{20432235459}a^{8}+\frac{94932694670}{61296706377}a^{7}+\frac{121725651209}{40864470918}a^{6}-\frac{21218464522}{6810745153}a^{5}+\frac{8996680450}{6810745153}a^{4}+\frac{11114772255}{6810745153}a^{3}+\frac{40551085455}{13621490306}a^{2}+\frac{102061454459}{40864470918}a+\frac{10614609659}{13621490306}$, $\frac{3607225741}{122593412754}a^{17}-\frac{9958972637}{122593412754}a^{16}+\frac{11009701691}{61296706377}a^{15}+\frac{11639061433}{40864470918}a^{14}-\frac{55462230998}{61296706377}a^{13}+\frac{80733687601}{40864470918}a^{12}-\frac{21773155855}{122593412754}a^{11}-\frac{84444104621}{61296706377}a^{10}+\frac{687679962341}{122593412754}a^{9}-\frac{320711754689}{40864470918}a^{8}+\frac{1130558716237}{122593412754}a^{7}-\frac{160547989909}{20432235459}a^{6}+\frac{163079549747}{40864470918}a^{5}-\frac{144727841597}{40864470918}a^{4}+\frac{27497050825}{13621490306}a^{3}-\frac{722451068}{6810745153}a^{2}+\frac{15597192847}{20432235459}a-\frac{19484723}{6810745153}$, $\frac{3315257591}{122593412754}a^{17}-\frac{10035141931}{122593412754}a^{16}+\frac{21528898867}{122593412754}a^{15}+\frac{5560329950}{20432235459}a^{14}-\frac{126325169575}{122593412754}a^{13}+\frac{42213383459}{20432235459}a^{12}+\frac{3595180669}{61296706377}a^{11}-\frac{346499310199}{122593412754}a^{10}+\frac{830700529991}{122593412754}a^{9}-\frac{796292648621}{122593412754}a^{8}+\frac{347562811960}{61296706377}a^{7}-\frac{136052215345}{40864470918}a^{6}+\frac{32465483765}{40864470918}a^{5}+\frac{17478802355}{13621490306}a^{4}+\frac{16768247797}{20432235459}a^{3}+\frac{11688713393}{40864470918}a^{2}+\frac{2645424356}{20432235459}a+\frac{2455021134}{6810745153}$, $\frac{3908730971}{122593412754}a^{17}-\frac{9815409899}{122593412754}a^{16}+\frac{18516421735}{122593412754}a^{15}+\frac{8971936814}{20432235459}a^{14}-\frac{43154914021}{40864470918}a^{13}+\frac{34771951325}{20432235459}a^{12}+\frac{90243567265}{61296706377}a^{11}-\frac{399687576413}{122593412754}a^{10}+\frac{715167249875}{122593412754}a^{9}-\frac{399619822805}{122593412754}a^{8}+\frac{173061804040}{61296706377}a^{7}+\frac{9409055539}{40864470918}a^{6}-\frac{60820024807}{40864470918}a^{5}+\frac{8737561195}{40864470918}a^{4}+\frac{32993738173}{20432235459}a^{3}+\frac{46233188885}{40864470918}a^{2}-\frac{1228717861}{20432235459}a+\frac{471213056}{6810745153}$, $\frac{727217338}{20432235459}a^{17}-\frac{3398538913}{40864470918}a^{16}+\frac{1714661491}{13621490306}a^{15}+\frac{3937871844}{6810745153}a^{14}-\frac{24784029425}{20432235459}a^{13}+\frac{8852142155}{6810745153}a^{12}+\frac{18531899824}{6810745153}a^{11}-\frac{64810619319}{13621490306}a^{10}+\frac{181643484635}{40864470918}a^{9}-\frac{8964882805}{40864470918}a^{8}-\frac{37479590105}{13621490306}a^{7}+\frac{116026635887}{40864470918}a^{6}-\frac{43293455019}{13621490306}a^{5}-\frac{9512682697}{13621490306}a^{4}+\frac{54628557243}{13621490306}a^{3}-\frac{7139467877}{13621490306}a^{2}-\frac{4942445361}{13621490306}a+\frac{4716574311}{6810745153}$, $\frac{1718540516}{61296706377}a^{17}-\frac{1050197621}{13621490306}a^{16}+\frac{6760967153}{40864470918}a^{15}+\frac{36670402823}{122593412754}a^{14}-\frac{18743617535}{20432235459}a^{13}+\frac{76302111541}{40864470918}a^{12}+\frac{12503440568}{61296706377}a^{11}-\frac{13806145073}{6810745153}a^{10}+\frac{78375939221}{13621490306}a^{9}-\frac{85032731535}{13621490306}a^{8}+\frac{274644064345}{40864470918}a^{7}-\frac{97486053238}{20432235459}a^{6}+\frac{81755709347}{40864470918}a^{5}-\frac{9161493963}{13621490306}a^{4}+\frac{28612517285}{13621490306}a^{3}+\frac{7080050546}{6810745153}a^{2}+\frac{5612661735}{13621490306}a-\frac{14705643}{6810745153}$ 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:  \( 87439.9587358 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 87439.9587358 \cdot 1}{6\cdot\sqrt{261508830075000000000000}}\cr\approx \mathstrut & 0.434944987233 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 + 11*x^15 - 39*x^14 + 69*x^13 + 13*x^12 - 114*x^11 + 225*x^10 - 219*x^9 + 174*x^8 - 72*x^7 - 24*x^6 + 45*x^5 + 36*x^4 + 18*x^3 - 9*x^2 + 9)
 
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 - 3*x^17 + 6*x^16 + 11*x^15 - 39*x^14 + 69*x^13 + 13*x^12 - 114*x^11 + 225*x^10 - 219*x^9 + 174*x^8 - 72*x^7 - 24*x^6 + 45*x^5 + 36*x^4 + 18*x^3 - 9*x^2 + 9, 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 - 3*x^17 + 6*x^16 + 11*x^15 - 39*x^14 + 69*x^13 + 13*x^12 - 114*x^11 + 225*x^10 - 219*x^9 + 174*x^8 - 72*x^7 - 24*x^6 + 45*x^5 + 36*x^4 + 18*x^3 - 9*x^2 + 9);
 
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 - 3*x^17 + 6*x^16 + 11*x^15 - 39*x^14 + 69*x^13 + 13*x^12 - 114*x^11 + 225*x^10 - 219*x^9 + 174*x^8 - 72*x^7 - 24*x^6 + 45*x^5 + 36*x^4 + 18*x^3 - 9*x^2 + 9);
 
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

$S_3\times D_6$ (as 18T29):

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 72
The 18 conjugacy class representatives for $S_3\times D_6$
Character table for $S_3\times D_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.135.1, 3.3.2700.1, 6.0.54675.1, 6.0.21870000.1, 9.3.295245000000.1

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 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.0.262440000000000.4

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 ${\href{/padicField/7.2.0.1}{2} }^{8}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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}$
\(3\) Copy content Toggle raw display 3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.11$x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 792 x^{8} + 1728 x^{7} + 2918 x^{6} + 3684 x^{5} + 3156 x^{4} + 1376 x^{3} - 36 x^{2} - 168 x + 25$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
\(5\) Copy content Toggle raw display 5.6.4.1$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.12.10.1$x^{12} + 20 x^{7} + 10 x^{6} + 50 x^{2} + 100 x + 25$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$