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Properties

Label 18.2.326...264.1
Degree $18$
Signature $[2, 8]$
Discriminant $3.267\times 10^{21}$
Root discriminant \(15.68\)
Ramified primes $2,3,107$
Class number $1$
Class group trivial
Galois group $C_6^2:D_6$ (as 18T154)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 2*x^16 - 12*x^15 + 7*x^14 + 7*x^13 + 60*x^12 - 7*x^11 - 10*x^10 - 97*x^9 - 10*x^8 - 7*x^7 + 60*x^6 + 7*x^5 + 7*x^4 - 12*x^3 + 2*x^2 - x + 1)
 
Copy content gp:K = bnfinit(y^18 - y^17 + 2*y^16 - 12*y^15 + 7*y^14 + 7*y^13 + 60*y^12 - 7*y^11 - 10*y^10 - 97*y^9 - 10*y^8 - 7*y^7 + 60*y^6 + 7*y^5 + 7*y^4 - 12*y^3 + 2*y^2 - y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 + 2*x^16 - 12*x^15 + 7*x^14 + 7*x^13 + 60*x^12 - 7*x^11 - 10*x^10 - 97*x^9 - 10*x^8 - 7*x^7 + 60*x^6 + 7*x^5 + 7*x^4 - 12*x^3 + 2*x^2 - x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - x^17 + 2*x^16 - 12*x^15 + 7*x^14 + 7*x^13 + 60*x^12 - 7*x^11 - 10*x^10 - 97*x^9 - 10*x^8 - 7*x^7 + 60*x^6 + 7*x^5 + 7*x^4 - 12*x^3 + 2*x^2 - x + 1)
 

\( x^{18} - x^{17} + 2 x^{16} - 12 x^{15} + 7 x^{14} + 7 x^{13} + 60 x^{12} - 7 x^{11} - 10 x^{10} + \cdots + 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $18$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[2, 8]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(3266763321003968139264\) \(\medspace = 2^{12}\cdot 3^{12}\cdot 107^{6}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(15.68\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{2/3}3^{3/4}107^{1/2}\approx 37.4299712608451$
Ramified primes:   \(2\), \(3\), \(107\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{10}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{1}{9}a^{5}+\frac{1}{9}a^{4}+\frac{2}{9}a^{3}-\frac{2}{9}a^{2}+\frac{1}{3}a+\frac{1}{9}$, $\frac{1}{9}a^{11}+\frac{1}{9}a^{9}-\frac{1}{9}a^{8}+\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{1}{9}a^{5}+\frac{2}{9}a^{4}-\frac{2}{9}a^{3}+\frac{1}{3}a^{2}+\frac{1}{9}a$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{9}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{9}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{9}$, $\frac{1}{9}a^{13}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{4}{9}a^{4}-\frac{1}{9}a^{3}-\frac{2}{9}a^{2}-\frac{4}{9}a-\frac{2}{9}$, $\frac{1}{27}a^{14}+\frac{1}{27}a^{13}+\frac{1}{27}a^{12}+\frac{1}{27}a^{11}+\frac{1}{27}a^{10}+\frac{1}{27}a^{9}+\frac{1}{9}a^{7}+\frac{1}{27}a^{5}-\frac{2}{27}a^{4}-\frac{8}{27}a^{3}+\frac{1}{27}a^{2}-\frac{5}{27}a-\frac{8}{27}$, $\frac{1}{27}a^{15}-\frac{1}{27}a^{9}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{1}{27}a^{6}-\frac{1}{9}a^{5}-\frac{2}{9}a^{4}+\frac{1}{3}a^{3}-\frac{2}{9}a^{2}-\frac{1}{9}a+\frac{8}{27}$, $\frac{1}{81}a^{16}+\frac{1}{81}a^{14}-\frac{2}{81}a^{13}+\frac{4}{81}a^{12}+\frac{4}{81}a^{11}-\frac{1}{27}a^{10}+\frac{13}{81}a^{9}-\frac{4}{27}a^{8}+\frac{13}{81}a^{7}-\frac{4}{27}a^{6}-\frac{5}{81}a^{5}-\frac{32}{81}a^{4}+\frac{7}{81}a^{3}+\frac{37}{81}a^{2}+\frac{1}{9}a-\frac{26}{81}$, $\frac{1}{243}a^{17}+\frac{1}{243}a^{16}-\frac{2}{243}a^{15}+\frac{2}{243}a^{14}+\frac{5}{243}a^{13}+\frac{2}{243}a^{12}+\frac{13}{243}a^{11}-\frac{5}{243}a^{10}+\frac{25}{243}a^{9}-\frac{35}{243}a^{8}-\frac{35}{243}a^{7}+\frac{34}{243}a^{6}-\frac{106}{243}a^{5}-\frac{67}{243}a^{4}-\frac{70}{243}a^{3}-\frac{59}{243}a^{2}+\frac{40}{243}a-\frac{2}{243}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

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

Class group and class number

Ideal class group:  Trivial group, which has order $1$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}$, which has order $2$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $9$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{32}{81}a^{17}-\frac{16}{81}a^{16}+\frac{32}{81}a^{15}-\frac{320}{81}a^{14}-\frac{8}{81}a^{13}+\frac{544}{81}a^{12}+\frac{1760}{81}a^{11}+\frac{620}{81}a^{10}-\frac{1120}{81}a^{9}-\frac{2272}{81}a^{8}-\frac{1402}{81}a^{7}+\frac{608}{81}a^{6}+\frac{832}{81}a^{5}+\frac{595}{81}a^{4}-\frac{320}{81}a^{3}+\frac{32}{81}a^{2}+\frac{98}{81}a+\frac{32}{81}$, $a$, $\frac{1}{243}a^{17}+\frac{52}{243}a^{16}-\frac{2}{243}a^{15}+\frac{44}{243}a^{14}-\frac{538}{243}a^{13}-\frac{181}{243}a^{12}+\frac{748}{243}a^{11}+\frac{3640}{243}a^{10}+\frac{2245}{243}a^{9}-\frac{971}{243}a^{8}-\frac{5798}{243}a^{7}-\frac{4061}{243}a^{6}-\frac{694}{243}a^{5}+\frac{2909}{243}a^{4}+\frac{1925}{243}a^{3}+\frac{415}{243}a^{2}-\frac{266}{243}a-\frac{95}{243}$, $\frac{49}{243}a^{17}-\frac{98}{243}a^{16}+\frac{199}{243}a^{15}-\frac{751}{243}a^{14}+\frac{1025}{243}a^{13}-\frac{598}{243}a^{12}+\frac{3019}{243}a^{11}-\frac{2666}{243}a^{10}+\frac{2365}{243}a^{9}-\frac{5162}{243}a^{8}+\frac{2719}{243}a^{7}-\frac{3428}{243}a^{6}+\frac{2966}{243}a^{5}-\frac{1360}{243}a^{4}+\frac{1940}{243}a^{3}-\frac{284}{243}a^{2}+\frac{556}{243}a-\frac{218}{243}$, $\frac{32}{81}a^{17}-\frac{1}{9}a^{16}+\frac{41}{81}a^{15}-\frac{334}{81}a^{14}-\frac{52}{81}a^{13}+\frac{389}{81}a^{12}+\frac{679}{27}a^{11}+\frac{1145}{81}a^{10}-\frac{146}{27}a^{9}-\frac{3013}{81}a^{8}-\frac{806}{27}a^{7}-\frac{529}{81}a^{6}+\frac{1190}{81}a^{5}+\frac{1421}{81}a^{4}+\frac{311}{81}a^{3}+\frac{8}{9}a^{2}-\frac{103}{81}a$, $\frac{1}{81}a^{17}+\frac{38}{81}a^{16}-\frac{38}{81}a^{15}+\frac{23}{27}a^{14}-\frac{151}{27}a^{13}+\frac{10}{3}a^{12}+\frac{344}{81}a^{11}+\frac{2173}{81}a^{10}-\frac{94}{81}a^{9}-\frac{380}{81}a^{8}-\frac{3289}{81}a^{7}-\frac{968}{81}a^{6}-\frac{31}{9}a^{5}+\frac{547}{27}a^{4}+\frac{232}{27}a^{3}+\frac{323}{81}a^{2}-\frac{128}{81}a-\frac{7}{81}$, $\frac{2}{243}a^{17}-\frac{52}{243}a^{16}+\frac{5}{243}a^{15}-\frac{77}{243}a^{14}+\frac{496}{243}a^{13}+\frac{247}{243}a^{12}-\frac{568}{243}a^{11}-\frac{3169}{243}a^{10}-\frac{2497}{243}a^{9}+\frac{173}{243}a^{8}+\frac{4007}{243}a^{7}+\frac{4100}{243}a^{6}+\frac{1489}{243}a^{5}-\frac{1187}{243}a^{4}-\frac{1490}{243}a^{3}-\frac{631}{243}a^{2}+\frac{242}{243}a+\frac{95}{243}$, $\frac{1}{9}a^{17}+\frac{23}{81}a^{16}-\frac{4}{9}a^{15}-\frac{25}{81}a^{14}-\frac{346}{81}a^{13}+\frac{512}{81}a^{12}+\frac{701}{81}a^{11}+18a^{10}-\frac{1351}{81}a^{9}-\frac{419}{27}a^{8}-\frac{2203}{81}a^{7}+\frac{313}{27}a^{6}+\frac{1016}{81}a^{5}+\frac{1088}{81}a^{4}-\frac{319}{81}a^{3}-\frac{340}{81}a^{2}-\frac{13}{27}a+\frac{56}{81}$, $\frac{70}{243}a^{17}-\frac{119}{243}a^{16}+\frac{220}{243}a^{15}-\frac{949}{243}a^{14}+\frac{1097}{243}a^{13}-\frac{166}{243}a^{12}+\frac{3817}{243}a^{11}-\frac{2843}{243}a^{10}+\frac{1570}{243}a^{9}-\frac{5771}{243}a^{8}+\frac{2437}{243}a^{7}-\frac{2876}{243}a^{6}+\frac{3398}{243}a^{5}-\frac{1486}{243}a^{4}+\frac{1625}{243}a^{3}-\frac{89}{243}a^{2}+\frac{361}{243}a-\frac{266}{243}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 4440.92692884 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 4440.92692884 \cdot 1}{2\cdot\sqrt{3266763321003968139264}}\cr\approx \mathstrut & 0.377471124154 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 2*x^16 - 12*x^15 + 7*x^14 + 7*x^13 + 60*x^12 - 7*x^11 - 10*x^10 - 97*x^9 - 10*x^8 - 7*x^7 + 60*x^6 + 7*x^5 + 7*x^4 - 12*x^3 + 2*x^2 - 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^18 - x^17 + 2*x^16 - 12*x^15 + 7*x^14 + 7*x^13 + 60*x^12 - 7*x^11 - 10*x^10 - 97*x^9 - 10*x^8 - 7*x^7 + 60*x^6 + 7*x^5 + 7*x^4 - 12*x^3 + 2*x^2 - 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 + 2*x^16 - 12*x^15 + 7*x^14 + 7*x^13 + 60*x^12 - 7*x^11 - 10*x^10 - 97*x^9 - 10*x^8 - 7*x^7 + 60*x^6 + 7*x^5 + 7*x^4 - 12*x^3 + 2*x^2 - 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - x^17 + 2*x^16 - 12*x^15 + 7*x^14 + 7*x^13 + 60*x^12 - 7*x^11 - 10*x^10 - 97*x^9 - 10*x^8 - 7*x^7 + 60*x^6 + 7*x^5 + 7*x^4 - 12*x^3 + 2*x^2 - 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

$C_6^2:D_6$ (as 18T154):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_6^2:D_6$
Character table for $C_6^2:D_6$

Intermediate fields

3.3.321.1, 6.2.927369.1, 9.3.6350622912.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 18 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 18.2.1333975179684221380409462784.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.3.3.6a1.1$x^{9} + 3 x^{7} + 3 x^{6} + 3 x^{5} + 6 x^{4} + 4 x^{3} + 3 x^{2} + 3 x + 3$$3$$3$$6$$S_3\times C_3$$$[\ ]_{3}^{6}$$
2.3.3.6a1.1$x^{9} + 3 x^{7} + 3 x^{6} + 3 x^{5} + 6 x^{4} + 4 x^{3} + 3 x^{2} + 3 x + 3$$3$$3$$6$$S_3\times C_3$$$[\ ]_{3}^{6}$$
\(3\) Copy content Toggle raw display 3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.3.4.9a1.2$x^{12} + 8 x^{10} + 4 x^{9} + 24 x^{8} + 24 x^{7} + 38 x^{6} + 48 x^{5} + 40 x^{4} + 36 x^{3} + 24 x^{2} + 8 x + 4$$4$$3$$9$$D_4 \times C_3$$$[\ ]_{4}^{6}$$
\(107\) Copy content Toggle raw display $\Q_{107}$$x + 105$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{107}$$x + 105$$1$$1$$0$Trivial$$[\ ]$$
107.2.1.0a1.1$x^{2} + 103 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
107.2.1.0a1.1$x^{2} + 103 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
107.2.2.2a1.2$x^{4} + 206 x^{3} + 10613 x^{2} + 412 x + 111$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
107.2.2.2a1.2$x^{4} + 206 x^{3} + 10613 x^{2} + 412 x + 111$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
107.2.2.2a1.2$x^{4} + 206 x^{3} + 10613 x^{2} + 412 x + 111$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)