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Properties

Label 12.12.989...125.1
Degree $12$
Signature $[12, 0]$
Discriminant $9.891\times 10^{18}$
Root discriminant \(38.28\)
Ramified primes $3,5,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{12}$ (as 12T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 36*x^10 + 106*x^9 + 393*x^8 - 1164*x^7 - 1350*x^6 + 4794*x^5 + 441*x^4 - 6643*x^3 + 1926*x^2 + 2865*x - 1349)
 
Copy content gp:K = bnfinit(y^12 - 3*y^11 - 36*y^10 + 106*y^9 + 393*y^8 - 1164*y^7 - 1350*y^6 + 4794*y^5 + 441*y^4 - 6643*y^3 + 1926*y^2 + 2865*y - 1349, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 - 36*x^10 + 106*x^9 + 393*x^8 - 1164*x^7 - 1350*x^6 + 4794*x^5 + 441*x^4 - 6643*x^3 + 1926*x^2 + 2865*x - 1349);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^11 - 36*x^10 + 106*x^9 + 393*x^8 - 1164*x^7 - 1350*x^6 + 4794*x^5 + 441*x^4 - 6643*x^3 + 1926*x^2 + 2865*x - 1349)
 

\( x^{12} - 3 x^{11} - 36 x^{10} + 106 x^{9} + 393 x^{8} - 1164 x^{7} - 1350 x^{6} + 4794 x^{5} + \cdots - 1349 \) 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:  $12$
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:  $[12, 0]$
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:   \(9891413435408203125\) \(\medspace = 3^{16}\cdot 5^{9}\cdot 7^{6}\) 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:  \(38.28\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{4/3}5^{3/4}7^{1/2}\approx 38.27702679912765$
Ramified primes:   \(3\), \(5\), \(7\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{12}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(64,·)$, $\chi_{315}(1,·)$, $\chi_{315}(211,·)$, $\chi_{315}(97,·)$, $\chi_{315}(169,·)$, $\chi_{315}(202,·)$, $\chi_{315}(13,·)$, $\chi_{315}(274,·)$, $\chi_{315}(307,·)$, $\chi_{315}(118,·)$, $\chi_{315}(106,·)$, $\chi_{315}(223,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19}a^{9}+\frac{4}{19}a^{8}+\frac{1}{19}a^{7}+\frac{8}{19}a^{6}+\frac{8}{19}a^{5}+\frac{1}{19}a^{4}-\frac{5}{19}a^{3}-\frac{8}{19}a^{2}+\frac{9}{19}a$, $\frac{1}{19}a^{10}+\frac{4}{19}a^{8}+\frac{4}{19}a^{7}-\frac{5}{19}a^{6}+\frac{7}{19}a^{5}-\frac{9}{19}a^{4}-\frac{7}{19}a^{3}+\frac{3}{19}a^{2}+\frac{2}{19}a$, $\frac{1}{1045156909429}a^{11}-\frac{2242936008}{1045156909429}a^{10}-\frac{24144692793}{1045156909429}a^{9}+\frac{33669755457}{1045156909429}a^{8}+\frac{316103098293}{1045156909429}a^{7}+\frac{99583078814}{1045156909429}a^{6}-\frac{5713060868}{11743336061}a^{5}-\frac{443185672267}{1045156909429}a^{4}+\frac{71685496487}{1045156909429}a^{3}+\frac{420826743567}{1045156909429}a^{2}-\frac{234702917662}{1045156909429}a+\frac{20409816750}{55008258391}$ 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 $1$ (assuming GRH)
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 $2$ (assuming GRH)
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:  $11$
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:   $\frac{11907}{2045521}a^{11}-\frac{30819}{2045521}a^{10}-\frac{456557}{2045521}a^{9}+\frac{1094634}{2045521}a^{8}+\frac{5694762}{2045521}a^{7}-\frac{12163826}{2045521}a^{6}-\frac{27520701}{2045521}a^{5}+\frac{51702033}{2045521}a^{4}+\frac{50198830}{2045521}a^{3}-\frac{79273935}{2045521}a^{2}-\frac{29759274}{2045521}a+\frac{1997934}{107659}$, $\frac{3604572825}{1045156909429}a^{11}-\frac{132880481}{55008258391}a^{10}-\frac{148199930155}{1045156909429}a^{9}+\frac{76839931740}{1045156909429}a^{8}+\frac{2036505254960}{1045156909429}a^{7}-\frac{717988514170}{1045156909429}a^{6}-\frac{124535807451}{11743336061}a^{5}+\frac{3955478826560}{1045156909429}a^{4}+\frac{23776442709100}{1045156909429}a^{3}-\frac{11032674479345}{1045156909429}a^{2}-\frac{14217501004015}{1045156909429}a+\frac{439233737907}{55008258391}$, $\frac{3604572825}{1045156909429}a^{11}-\frac{132880481}{55008258391}a^{10}-\frac{148199930155}{1045156909429}a^{9}+\frac{76839931740}{1045156909429}a^{8}+\frac{2036505254960}{1045156909429}a^{7}-\frac{717988514170}{1045156909429}a^{6}-\frac{124535807451}{11743336061}a^{5}+\frac{3955478826560}{1045156909429}a^{4}+\frac{23776442709100}{1045156909429}a^{3}-\frac{11032674479345}{1045156909429}a^{2}-\frac{14217501004015}{1045156909429}a+\frac{384225479516}{55008258391}$, $\frac{251832218}{55008258391}a^{11}-\frac{10300838629}{1045156909429}a^{10}-\frac{186879209874}{1045156909429}a^{9}+\frac{353799003752}{1045156909429}a^{8}+\frac{2403780332209}{1045156909429}a^{7}-\frac{3670615391235}{1045156909429}a^{6}-\frac{136735027706}{11743336061}a^{5}+\frac{13629186519807}{1045156909429}a^{4}+\frac{23187183661440}{1045156909429}a^{3}-\frac{15679241004492}{1045156909429}a^{2}-\frac{647538098810}{55008258391}a+\frac{328092812039}{55008258391}$, $\frac{29210587230}{1045156909429}a^{11}-\frac{67652199719}{1045156909429}a^{10}-\frac{1091939638094}{1045156909429}a^{9}+\frac{2327886531933}{1045156909429}a^{8}+\frac{12875475712758}{1045156909429}a^{7}-\frac{24448087064585}{1045156909429}a^{6}-\frac{610584948585}{11743336061}a^{5}+\frac{95136321659787}{1045156909429}a^{4}+\frac{74322030269016}{1045156909429}a^{3}-\frac{116505524485975}{1045156909429}a^{2}-\frac{31467810678154}{1045156909429}a+\frac{2123218762305}{55008258391}$, $\frac{3718638927}{1045156909429}a^{11}-\frac{282256083}{1045156909429}a^{10}-\frac{147345877217}{1045156909429}a^{9}-\frac{23922657801}{1045156909429}a^{8}+\frac{1883520078282}{1045156909429}a^{7}+\frac{802749626184}{1045156909429}a^{6}-\frac{99133422651}{11743336061}a^{5}-\frac{4991402659572}{1045156909429}a^{4}+\frac{15263429647440}{1045156909429}a^{3}+\frac{10630393017660}{1045156909429}a^{2}-\frac{8679476581308}{1045156909429}a-\frac{349612057236}{55008258391}$, $\frac{35319832053}{1045156909429}a^{11}-\frac{84434960703}{1045156909429}a^{10}-\frac{1325727613341}{1045156909429}a^{9}+\frac{2923498603812}{1045156909429}a^{8}+\frac{15778510617696}{1045156909429}a^{7}-\frac{31014526080954}{1045156909429}a^{6}-\frac{766008053661}{11743336061}a^{5}+\frac{6424097679966}{55008258391}a^{4}+\frac{98008681866550}{1045156909429}a^{3}-\frac{154461203381364}{1045156909429}a^{2}-\frac{44764750197339}{1045156909429}a+\frac{2994407251965}{55008258391}$, $\frac{15622399366}{1045156909429}a^{11}-\frac{22101358371}{1045156909429}a^{10}-\frac{597958696371}{1045156909429}a^{9}+\frac{37552590467}{55008258391}a^{8}+\frac{7285554742584}{1045156909429}a^{7}-\frac{6814448403570}{1045156909429}a^{6}-\frac{360163325950}{11743336061}a^{5}+\frac{25967494266789}{1045156909429}a^{4}+\frac{48828300883818}{1045156909429}a^{3}-\frac{32732367001119}{1045156909429}a^{2}-\frac{21963873701197}{1045156909429}a+\frac{736939629057}{55008258391}$, $\frac{66329694413}{1045156909429}a^{11}-\frac{200902424471}{1045156909429}a^{10}-\frac{2341296207875}{1045156909429}a^{9}+\frac{6953722511280}{1045156909429}a^{8}+\frac{24465995118315}{1045156909429}a^{7}-\frac{72866787907915}{1045156909429}a^{6}-\frac{827658998152}{11743336061}a^{5}+\frac{266868616404356}{1045156909429}a^{4}-\frac{11476162102950}{1045156909429}a^{3}-\frac{248308272685383}{1045156909429}a^{2}+\frac{83429468760477}{1045156909429}a+\frac{1252071764481}{55008258391}$, $\frac{108237957450}{1045156909429}a^{11}-\frac{11832723557}{55008258391}a^{10}-\frac{4109127951103}{1045156909429}a^{9}+\frac{7696763139963}{1045156909429}a^{8}+\frac{2621349493031}{55008258391}a^{7}-\frac{80595277109816}{1045156909429}a^{6}-\frac{2493620734873}{11743336061}a^{5}+\frac{320333380291695}{1045156909429}a^{4}+\frac{345049411852922}{1045156909429}a^{3}-\frac{22263873407848}{55008258391}a^{2}-\frac{169404883324736}{1045156909429}a+\frac{8936505707850}{55008258391}$, $\frac{217717789388}{1045156909429}a^{11}-\frac{417272433938}{1045156909429}a^{10}-\frac{8293146598225}{1045156909429}a^{9}+\frac{14094373437874}{1045156909429}a^{8}+\frac{100952130791220}{1045156909429}a^{7}-\frac{144069820737013}{1045156909429}a^{6}-\frac{5070905779424}{11743336061}a^{5}+\frac{554691523327267}{1045156909429}a^{4}+\frac{700923820280822}{1045156909429}a^{3}-\frac{686944750664067}{1045156909429}a^{2}-\frac{326064108878025}{1045156909429}a+\frac{14323875602157}{55008258391}$ Copy content Toggle raw display (assuming GRH)
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:  \( 413348.51768 \) (assuming GRH)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 413348.51768 \cdot 1}{2\cdot\sqrt{9891413435408203125}}\cr\approx \mathstrut & 0.26916411520 \end{aligned}\] (assuming GRH)

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^12 - 3*x^11 - 36*x^10 + 106*x^9 + 393*x^8 - 1164*x^7 - 1350*x^6 + 4794*x^5 + 441*x^4 - 6643*x^3 + 1926*x^2 + 2865*x - 1349) 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^12 - 3*x^11 - 36*x^10 + 106*x^9 + 393*x^8 - 1164*x^7 - 1350*x^6 + 4794*x^5 + 441*x^4 - 6643*x^3 + 1926*x^2 + 2865*x - 1349, 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^12 - 3*x^11 - 36*x^10 + 106*x^9 + 393*x^8 - 1164*x^7 - 1350*x^6 + 4794*x^5 + 441*x^4 - 6643*x^3 + 1926*x^2 + 2865*x - 1349); 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 = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 - 36*x^10 + 106*x^9 + 393*x^8 - 1164*x^7 - 1350*x^6 + 4794*x^5 + 441*x^4 - 6643*x^3 + 1926*x^2 + 2865*x - 1349); 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

$C_{12}$ (as 12T1):

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); G, transitive_group_identification(G)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.4.6125.1, 6.6.820125.1

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.12.0.1}{12} }$ R R R ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.4.0.1}{4} }^{3}$ ${\href{/padicField/19.1.0.1}{1} }^{12}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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
\(3\) Copy content Toggle raw display 3.4.3.16a2.1$x^{12} + 6 x^{11} + 12 x^{10} + 8 x^{9} + 12 x^{8} + 48 x^{7} + 48 x^{6} + 36 x^{4} + 72 x^{3} + 35$$3$$4$$16$$C_{12}$$$[2]^{4}$$
\(5\) Copy content Toggle raw display 5.3.4.9a1.1$x^{12} + 12 x^{10} + 12 x^{9} + 54 x^{8} + 108 x^{7} + 162 x^{6} + 324 x^{5} + 405 x^{4} + 432 x^{3} + 491 x^{2} + 324 x + 81$$4$$3$$9$$C_{12}$$$[\ ]_{4}^{3}$$
\(7\) Copy content Toggle raw display 7.6.2.6a1.1$x^{12} + 2 x^{10} + 10 x^{9} + 9 x^{8} + 22 x^{7} + 39 x^{6} + 52 x^{5} + 82 x^{4} + 78 x^{3} + 60 x^{2} + 43 x + 9$$2$$6$$6$$C_{12}$$$[\ ]_{2}^{6}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*12 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*12 1.35.4t1.a.a$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
*12 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
*12 1.35.4t1.a.b$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
*12 1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
*12 1.315.12t1.a.a$1$ $ 3^{2} \cdot 5 \cdot 7 $ 12.12.9891413435408203125.1 $C_{12}$ (as 12T1) $0$ $1$
*12 1.45.6t1.a.a$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
*12 1.315.12t1.a.b$1$ $ 3^{2} \cdot 5 \cdot 7 $ 12.12.9891413435408203125.1 $C_{12}$ (as 12T1) $0$ $1$
*12 1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
*12 1.315.12t1.a.c$1$ $ 3^{2} \cdot 5 \cdot 7 $ 12.12.9891413435408203125.1 $C_{12}$ (as 12T1) $0$ $1$
*12 1.45.6t1.a.b$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
*12 1.315.12t1.a.d$1$ $ 3^{2} \cdot 5 \cdot 7 $ 12.12.9891413435408203125.1 $C_{12}$ (as 12T1) $0$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers

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