Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 12*x^16 - 6*x^15 + 108*x^14 - 42*x^13 + 381*x^12 - 36*x^11 + 948*x^10 - 38*x^9 + 972*x^8 + 186*x^7 + 690*x^6 + 108*x^5 + 186*x^4 + 60*x^3 + 36*x^2 + 6*x + 1)
gp: K = bnfinit(y^18 + 12*y^16 - 6*y^15 + 108*y^14 - 42*y^13 + 381*y^12 - 36*y^11 + 948*y^10 - 38*y^9 + 972*y^8 + 186*y^7 + 690*y^6 + 108*y^5 + 186*y^4 + 60*y^3 + 36*y^2 + 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 12*x^16 - 6*x^15 + 108*x^14 - 42*x^13 + 381*x^12 - 36*x^11 + 948*x^10 - 38*x^9 + 972*x^8 + 186*x^7 + 690*x^6 + 108*x^5 + 186*x^4 + 60*x^3 + 36*x^2 + 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 12*x^16 - 6*x^15 + 108*x^14 - 42*x^13 + 381*x^12 - 36*x^11 + 948*x^10 - 38*x^9 + 972*x^8 + 186*x^7 + 690*x^6 + 108*x^5 + 186*x^4 + 60*x^3 + 36*x^2 + 6*x + 1)
\( x^{18} + 12 x^{16} - 6 x^{15} + 108 x^{14} - 42 x^{13} + 381 x^{12} - 36 x^{11} + 948 x^{10} - 38 x^{9} + \cdots + 1 \)
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: |
\(-91437830330543390573883\)
\(\medspace = -\,3^{31}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.86\) | 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: | $3^{31/18}23^{1/2}\approx 31.810664815402767$ | ||
| Ramified primes: |
\(3\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{18}a^{9}+\frac{1}{3}a^{7}-\frac{1}{6}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{18}$, $\frac{1}{18}a^{10}+\frac{1}{3}a^{8}-\frac{1}{6}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{18}a$, $\frac{1}{18}a^{11}-\frac{1}{6}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{18}a^{2}+\frac{1}{3}$, $\frac{1}{90}a^{12}-\frac{1}{90}a^{11}+\frac{1}{45}a^{10}+\frac{1}{6}a^{8}+\frac{2}{5}a^{7}-\frac{1}{10}a^{6}-\frac{2}{15}a^{5}-\frac{2}{5}a^{4}-\frac{7}{90}a^{3}-\frac{23}{90}a^{2}-\frac{7}{45}a-\frac{1}{10}$, $\frac{1}{90}a^{13}+\frac{1}{90}a^{11}+\frac{1}{45}a^{10}-\frac{13}{30}a^{8}+\frac{3}{10}a^{7}+\frac{4}{15}a^{6}+\frac{7}{15}a^{5}-\frac{43}{90}a^{4}-\frac{1}{3}a^{3}-\frac{37}{90}a^{2}-\frac{23}{90}a+\frac{1}{15}$, $\frac{1}{90}a^{14}-\frac{1}{45}a^{11}-\frac{1}{45}a^{10}+\frac{1}{90}a^{9}+\frac{3}{10}a^{8}-\frac{7}{15}a^{7}-\frac{1}{10}a^{6}+\frac{29}{90}a^{5}-\frac{4}{15}a^{4}+\frac{1}{18}a^{2}-\frac{1}{9}a+\frac{29}{90}$, $\frac{1}{90}a^{15}+\frac{1}{90}a^{11}+\frac{1}{45}a^{9}+\frac{11}{30}a^{8}+\frac{1}{5}a^{7}+\frac{13}{45}a^{6}+\frac{7}{15}a^{5}+\frac{1}{5}a^{4}-\frac{13}{30}a^{3}-\frac{1}{90}a^{2}+\frac{2}{5}a+\frac{37}{90}$, $\frac{1}{90}a^{16}+\frac{1}{90}a^{11}-\frac{1}{45}a^{9}+\frac{1}{30}a^{8}-\frac{4}{9}a^{7}-\frac{4}{15}a^{6}+\frac{1}{3}a^{5}-\frac{11}{30}a^{4}-\frac{4}{15}a^{3}-\frac{31}{90}a^{2}+\frac{7}{30}a+\frac{22}{45}$, $\frac{1}{24900074190}a^{17}-\frac{34778443}{12450037095}a^{16}-\frac{108660017}{24900074190}a^{15}+\frac{47970338}{12450037095}a^{14}+\frac{36633937}{24900074190}a^{13}-\frac{63713179}{12450037095}a^{12}+\frac{601612531}{24900074190}a^{11}-\frac{173427214}{12450037095}a^{10}-\frac{341827384}{12450037095}a^{9}+\frac{718703329}{4980014838}a^{8}-\frac{4665652747}{24900074190}a^{7}-\frac{2961500654}{12450037095}a^{6}-\frac{6146275831}{24900074190}a^{5}-\frac{9222550543}{24900074190}a^{4}+\frac{4567698149}{24900074190}a^{3}+\frac{3564314476}{12450037095}a^{2}-\frac{6271822789}{24900074190}a-\frac{1802969611}{24900074190}$
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: |
\( \frac{35944573}{8300024730} a^{17} + \frac{137885662}{1383337455} a^{16} + \frac{34176001}{1383337455} a^{15} + \frac{4875268667}{4150012365} a^{14} - \frac{1867536614}{4150012365} a^{13} + \frac{44814452569}{4150012365} a^{12} - \frac{44986232093}{8300024730} a^{11} + \frac{32681382094}{830002473} a^{10} - \frac{38387839457}{4150012365} a^{9} + \frac{799759780829}{8300024730} a^{8} - \frac{10473736256}{461112485} a^{7} + \frac{9348225913}{92222497} a^{6} + \frac{13290035483}{8300024730} a^{5} + \frac{269043101594}{4150012365} a^{4} - \frac{278781896}{830002473} a^{3} + \frac{68616360328}{4150012365} a^{2} + \frac{19761206962}{4150012365} a + \frac{7640711774}{4150012365} \)
(order $18$)
| 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: |
$\frac{340854686}{1383337455}a^{17}-\frac{12776186}{830002473}a^{16}+\frac{12080657741}{4150012365}a^{15}-\frac{2728914607}{1660004946}a^{14}+\frac{21712214060}{830002473}a^{13}-\frac{9550474255}{830002473}a^{12}+\frac{371536637734}{4150012365}a^{11}-\frac{14912054824}{1383337455}a^{10}+\frac{898322090041}{4150012365}a^{9}-\frac{42367200253}{2766674910}a^{8}+\frac{821899749478}{4150012365}a^{7}+\frac{207900880018}{4150012365}a^{6}+\frac{1054564940933}{8300024730}a^{5}+\frac{101769103544}{4150012365}a^{4}+\frac{87991084769}{4150012365}a^{3}+\frac{158828967697}{8300024730}a^{2}+\frac{426307712}{92222497}a+\frac{1734128054}{4150012365}$, $\frac{182858709}{922224970}a^{17}+\frac{92320496}{830002473}a^{16}+\frac{1947912703}{830002473}a^{15}+\frac{1182878041}{8300024730}a^{14}+\frac{28158781196}{1383337455}a^{13}+\frac{15960109099}{4150012365}a^{12}+\frac{279568347106}{4150012365}a^{11}+\frac{150621232556}{4150012365}a^{10}+\frac{712103768242}{4150012365}a^{9}+\frac{90164385997}{922224970}a^{8}+\frac{654151246214}{4150012365}a^{7}+\frac{590861205526}{4150012365}a^{6}+\frac{520867492042}{4150012365}a^{5}+\frac{39825776844}{461112485}a^{4}+\frac{21804518557}{830002473}a^{3}+\frac{98986821119}{4150012365}a^{2}+\frac{30352378204}{4150012365}a+\frac{10655108876}{4150012365}$, $\frac{438579406}{4150012365}a^{17}+\frac{12804460}{830002473}a^{16}+\frac{5235387638}{4150012365}a^{15}-\frac{616062004}{1383337455}a^{14}+\frac{46630926226}{4150012365}a^{13}-\frac{22341696763}{8300024730}a^{12}+\frac{161106171368}{4150012365}a^{11}+\frac{11388490789}{4150012365}a^{10}+\frac{133440200894}{1383337455}a^{9}+\frac{50475887486}{4150012365}a^{8}+\frac{391092213676}{4150012365}a^{7}+\frac{62342721157}{1660004946}a^{6}+\frac{30324290248}{461112485}a^{5}+\frac{18631388617}{830002473}a^{4}+\frac{126370161829}{8300024730}a^{3}+\frac{35801444266}{4150012365}a^{2}+\frac{11772858029}{4150012365}a+\frac{1134540371}{2766674910}$, $\frac{444872014}{4150012365}a^{17}-\frac{188052083}{4150012365}a^{16}+\frac{2054154797}{1660004946}a^{15}-\frac{3284435149}{2766674910}a^{14}+\frac{93648238169}{8300024730}a^{13}-\frac{75529841377}{8300024730}a^{12}+\frac{52163651387}{1383337455}a^{11}-\frac{52994002583}{2766674910}a^{10}+\frac{720437339119}{8300024730}a^{9}-\frac{378938250323}{8300024730}a^{8}+\frac{55507776541}{830002473}a^{7}-\frac{87403992296}{4150012365}a^{6}+\frac{99747575437}{2766674910}a^{5}-\frac{203766798893}{8300024730}a^{4}-\frac{12442068472}{4150012365}a^{3}+\frac{6332909447}{2766674910}a^{2}+\frac{1114233943}{461112485}a+\frac{3026194127}{8300024730}$, $\frac{319273718}{1383337455}a^{17}-\frac{5843991}{184444994}a^{16}+\frac{11366959711}{4150012365}a^{15}-\frac{14559968263}{8300024730}a^{14}+\frac{102684815641}{4150012365}a^{13}-\frac{35411570047}{2766674910}a^{12}+\frac{356169169162}{4150012365}a^{11}-\frac{148201314679}{8300024730}a^{10}+\frac{171933240961}{830002473}a^{9}-\frac{30435665677}{922224970}a^{8}+\frac{107203296301}{553334982}a^{7}+\frac{206241832453}{8300024730}a^{6}+\frac{195204993569}{1660004946}a^{5}+\frac{89390736649}{8300024730}a^{4}+\frac{14475929873}{922224970}a^{3}+\frac{103348228759}{8300024730}a^{2}+\frac{5023841177}{4150012365}a+\frac{993419053}{1660004946}$, $\frac{3121903}{553334982}a^{17}-\frac{39444607}{4150012365}a^{16}+\frac{716234249}{8300024730}a^{15}-\frac{1190611379}{8300024730}a^{14}+\frac{3700759817}{4150012365}a^{13}-\frac{1833420269}{1383337455}a^{12}+\frac{37811082791}{8300024730}a^{11}-\frac{17329678081}{4150012365}a^{10}+\frac{106512457147}{8300024730}a^{9}-\frac{11910145624}{1383337455}a^{8}+\frac{99899008477}{4150012365}a^{7}-\frac{44421490247}{8300024730}a^{6}+\frac{87703173484}{4150012365}a^{5}+\frac{1824900331}{4150012365}a^{4}+\frac{40128067561}{2766674910}a^{3}+\frac{32249696809}{8300024730}a^{2}+\frac{15238670332}{4150012365}a+\frac{2626698508}{4150012365}$, $\frac{5222916083}{8300024730}a^{17}-\frac{113781293}{4150012365}a^{16}+\frac{12374748131}{1660004946}a^{15}-\frac{5626579841}{1383337455}a^{14}+\frac{277942628542}{4150012365}a^{13}-\frac{78485776841}{2766674910}a^{12}+\frac{955889023943}{4150012365}a^{11}-\frac{104223922604}{4150012365}a^{10}+\frac{929482439023}{1660004946}a^{9}-\frac{136617644878}{4150012365}a^{8}+\frac{2167027721516}{4150012365}a^{7}+\frac{531077594614}{4150012365}a^{6}+\frac{937421832979}{2766674910}a^{5}+\frac{265062806792}{4150012365}a^{4}+\frac{78875775778}{1383337455}a^{3}+\frac{385952920109}{8300024730}a^{2}+\frac{30973511729}{4150012365}a+\frac{1916233913}{1660004946}$, $\frac{5464161721}{8300024730}a^{17}-\frac{1018661731}{8300024730}a^{16}+\frac{32523031567}{4150012365}a^{15}-\frac{14941242911}{2766674910}a^{14}+\frac{294948567947}{4150012365}a^{13}-\frac{55680979733}{1383337455}a^{12}+\frac{344382193852}{1383337455}a^{11}-\frac{180595656847}{2766674910}a^{10}+\frac{500328757856}{830002473}a^{9}-\frac{1075684456603}{8300024730}a^{8}+\frac{4827209541193}{8300024730}a^{7}+\frac{120790209176}{4150012365}a^{6}+\frac{504248452319}{1383337455}a^{5}+\frac{3236881807}{1660004946}a^{4}+\frac{18702692357}{276667491}a^{3}+\frac{12429635449}{461112485}a^{2}+\frac{3182965592}{461112485}a+\frac{2535819191}{4150012365}$
| 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: | \( 118867.593913 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 118867.593913 \cdot 1}{18\cdot\sqrt{91437830330543390573883}}\cr\approx \mathstrut & 0.333308835151 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 12*x^16 - 6*x^15 + 108*x^14 - 42*x^13 + 381*x^12 - 36*x^11 + 948*x^10 - 38*x^9 + 972*x^8 + 186*x^7 + 690*x^6 + 108*x^5 + 186*x^4 + 60*x^3 + 36*x^2 + 6*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 + 12*x^16 - 6*x^15 + 108*x^14 - 42*x^13 + 381*x^12 - 36*x^11 + 948*x^10 - 38*x^9 + 972*x^8 + 186*x^7 + 690*x^6 + 108*x^5 + 186*x^4 + 60*x^3 + 36*x^2 + 6*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 + 12*x^16 - 6*x^15 + 108*x^14 - 42*x^13 + 381*x^12 - 36*x^11 + 948*x^10 - 38*x^9 + 972*x^8 + 186*x^7 + 690*x^6 + 108*x^5 + 186*x^4 + 60*x^3 + 36*x^2 + 6*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 + 12*x^16 - 6*x^15 + 108*x^14 - 42*x^13 + 381*x^12 - 36*x^11 + 948*x^10 - 38*x^9 + 972*x^8 + 186*x^7 + 690*x^6 + 108*x^5 + 186*x^4 + 60*x^3 + 36*x^2 + 6*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
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 3.3.621.1, \(\Q(\zeta_{9})\), 6.0.1156923.1, 9.9.174583151469.1 |
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: | 12.0.57352136505929721.2 |
| Degree 18 sibling: | deg 18 |
| Minimal sibling: | 12.0.57352136505929721.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $18$ | $1$ | $31$ | |||
|
\(23\)
| 23.6.0.1 | $x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 23.12.6.1 | $x^{12} + 140 x^{10} + 18 x^{9} + 8092 x^{8} + 20 x^{7} + 244001 x^{6} - 56140 x^{5} + 4059563 x^{4} - 1721304 x^{3} + 36312468 x^{2} - 14694000 x + 141161628$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |