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Properties

Label 24.0.385...816.1
Degree $24$
Signature $(0, 12)$
Discriminant $3.852\times 10^{30}$
Root discriminant \(18.81\)
Ramified primes $2,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_3\times C_{12}$ (as 24T65)

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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^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 1)
 
Copy content gp:K = bnfinit(y^24 - 3*y^16 - 8*y^12 + 18*y^8 + 8*y^4 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 1)
 

\( x^{24} - 3x^{16} - 8x^{12} + 18x^{8} + 8x^{4} + 1 \) 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:  $24$
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:  $(0, 12)$
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:   \(3852179415897489839182437154816\) \(\medspace = 2^{72}\cdot 13^{8}\) 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:  \(18.81\)
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:  $2^{3}13^{2/3}\approx 44.23019850943098$
Ramified primes:   \(2\), \(13\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_{12}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\zeta_{16})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{899}a^{20}+\frac{287}{899}a^{16}-\frac{342}{899}a^{12}-\frac{171}{899}a^{8}+\frac{386}{899}a^{4}+\frac{213}{899}$, $\frac{1}{899}a^{21}+\frac{287}{899}a^{17}-\frac{342}{899}a^{13}-\frac{171}{899}a^{9}+\frac{386}{899}a^{5}+\frac{213}{899}a$, $\frac{1}{899}a^{22}+\frac{287}{899}a^{18}-\frac{342}{899}a^{14}-\frac{171}{899}a^{10}+\frac{386}{899}a^{6}+\frac{213}{899}a^{2}$, $\frac{1}{899}a^{23}+\frac{287}{899}a^{19}-\frac{342}{899}a^{15}-\frac{171}{899}a^{11}+\frac{386}{899}a^{7}+\frac{213}{899}a^{3}$ 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:  Trivial group, which has order $1$ (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:   \( \frac{952}{899} a^{23} - \frac{72}{899} a^{19} - \frac{2843}{899} a^{15} - \frac{7265}{899} a^{11} + \frac{17761}{899} a^{7} + \frac{5895}{899} a^{3} \)  (order $16$) 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:   $\frac{313}{899}a^{20}-\frac{69}{899}a^{16}-\frac{964}{899}a^{12}-\frac{2280}{899}a^{8}+\frac{5746}{899}a^{4}+\frac{1042}{899}$, $\frac{313}{899}a^{20}-\frac{69}{899}a^{16}-\frac{964}{899}a^{12}-\frac{2280}{899}a^{8}+\frac{6645}{899}a^{4}+\frac{1042}{899}$, $\frac{989}{899}a^{23}+\frac{573}{899}a^{21}-\frac{241}{899}a^{19}-\frac{66}{899}a^{17}-\frac{2911}{899}a^{15}-\frac{1782}{899}a^{13}-\frac{7299}{899}a^{11}-\frac{4487}{899}a^{9}+\frac{19457}{899}a^{7}+\frac{10812}{899}a^{5}+\frac{2988}{899}a^{3}+\frac{3381}{899}a$, $\frac{1042}{899}a^{22}+\frac{138}{899}a^{20}-\frac{313}{899}a^{18}+\frac{50}{899}a^{16}-\frac{3057}{899}a^{14}-\frac{448}{899}a^{12}-\frac{7372}{899}a^{10}-\frac{1123}{899}a^{8}+\frac{21036}{899}a^{6}+\frac{2025}{899}a^{4}+\frac{2590}{899}a^{2}+\frac{1525}{899}$, $\frac{53}{899}a^{22}-\frac{313}{899}a^{20}-\frac{72}{899}a^{18}+\frac{69}{899}a^{16}-\frac{146}{899}a^{14}+\frac{964}{899}a^{12}-\frac{73}{899}a^{10}+\frac{2280}{899}a^{8}+\frac{1579}{899}a^{6}-\frac{5746}{899}a^{4}-\frac{1297}{899}a^{2}-\frac{1042}{899}$, $\frac{814}{899}a^{22}+\frac{626}{899}a^{21}+\frac{15}{31}a^{20}-\frac{122}{899}a^{18}-\frac{138}{899}a^{17}-\frac{4}{31}a^{16}-\frac{2395}{899}a^{14}-\frac{1928}{899}a^{13}-\frac{46}{31}a^{12}-\frac{6142}{899}a^{10}-\frac{4560}{899}a^{9}-\frac{116}{31}a^{8}+\frac{15736}{899}a^{6}+\frac{12391}{899}a^{5}+\frac{303}{31}a^{4}+\frac{4370}{899}a^{2}+\frac{2983}{899}a+\frac{64}{31}$, $\frac{952}{899}a^{23}-\frac{488}{899}a^{22}-\frac{15}{31}a^{20}-\frac{72}{899}a^{19}+\frac{188}{899}a^{18}+\frac{4}{31}a^{16}-\frac{2843}{899}a^{15}+\frac{1480}{899}a^{14}+\frac{46}{31}a^{12}-\frac{7265}{899}a^{11}+\frac{3437}{899}a^{10}+\frac{116}{31}a^{8}+\frac{17761}{899}a^{7}-\frac{10366}{899}a^{6}-\frac{303}{31}a^{4}+\frac{5895}{899}a^{3}-\frac{559}{899}a^{2}-\frac{64}{31}$, $\frac{1127}{899}a^{23}-\frac{814}{899}a^{22}+\frac{501}{899}a^{21}+\frac{363}{899}a^{20}-\frac{191}{899}a^{19}+\frac{122}{899}a^{18}-\frac{53}{899}a^{17}-\frac{103}{899}a^{16}-\frac{3359}{899}a^{15}+\frac{2395}{899}a^{14}-\frac{1431}{899}a^{13}-\frac{983}{899}a^{12}-\frac{8422}{899}a^{11}+\frac{6142}{899}a^{10}-\frac{3862}{899}a^{9}-\frac{2739}{899}a^{8}+\frac{21482}{899}a^{7}-\frac{15736}{899}a^{6}+\frac{9091}{899}a^{5}+\frac{7066}{899}a^{4}+\frac{5412}{899}a^{3}-\frac{3471}{899}a^{2}+\frac{3328}{899}a+\frac{904}{899}$, $\frac{814}{899}a^{23}+\frac{53}{899}a^{21}+\frac{122}{899}a^{20}-\frac{122}{899}a^{19}-\frac{72}{899}a^{17}-\frac{47}{899}a^{16}-\frac{2395}{899}a^{15}-\frac{146}{899}a^{13}-\frac{370}{899}a^{12}-\frac{6142}{899}a^{11}-\frac{73}{899}a^{9}-\frac{1084}{899}a^{8}+\frac{15736}{899}a^{7}+\frac{1579}{899}a^{5}+\frac{2142}{899}a^{4}+\frac{4370}{899}a^{3}-a^{2}-\frac{398}{899}a-\frac{85}{899}$, $\frac{90}{899}a^{23}+\frac{1042}{899}a^{22}+\frac{814}{899}a^{21}+\frac{326}{899}a^{20}-\frac{241}{899}a^{19}-\frac{313}{899}a^{18}-\frac{122}{899}a^{17}+\frac{66}{899}a^{16}-\frac{214}{899}a^{15}-\frac{3057}{899}a^{14}-\frac{2395}{899}a^{13}-\frac{915}{899}a^{12}-\frac{107}{899}a^{11}-\frac{7372}{899}a^{10}-\frac{6142}{899}a^{9}-\frac{2705}{899}a^{8}+\frac{3275}{899}a^{7}+\frac{21036}{899}a^{6}+\frac{15736}{899}a^{5}+\frac{5370}{899}a^{4}-\frac{4204}{899}a^{3}+\frac{2590}{899}a^{2}+\frac{4370}{899}a+\frac{2912}{899}$, $\frac{573}{899}a^{23}+\frac{379}{899}a^{22}-\frac{379}{899}a^{21}+\frac{379}{899}a^{20}-\frac{66}{899}a^{19}-\frac{6}{899}a^{18}+\frac{6}{899}a^{17}-\frac{6}{899}a^{16}-\frac{1782}{899}a^{15}-\frac{1061}{899}a^{14}+\frac{1061}{899}a^{13}-\frac{1061}{899}a^{12}-\frac{4487}{899}a^{11}-\frac{2778}{899}a^{10}+\frac{2778}{899}a^{9}-\frac{2778}{899}a^{8}+\frac{10812}{899}a^{7}+\frac{6949}{899}a^{6}-\frac{6949}{899}a^{5}+\frac{6949}{899}a^{4}+\frac{4280}{899}a^{3}+\frac{1615}{899}a^{2}-\frac{1615}{899}a+\frac{1615}{899}$ 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:  \( 2446825.407448486 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 2446825.407448486 \cdot 1}{16\cdot\sqrt{3852179415897489839182437154816}}\cr\approx \mathstrut & 0.294977007875353 \end{aligned}\] (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^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 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^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 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^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 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^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 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

$S_3\times C_{12}$ (as 24T65):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
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 72
The 36 conjugacy class representatives for $S_3\times C_{12}$
Character table for $S_3\times C_{12}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), 4.0.2048.2, \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{8})\), 6.0.10816.1, \(\Q(\zeta_{16})\), 12.0.479174066176.4

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 36 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }^{2}$ ${\href{/padicField/5.12.0.1}{12} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{4}$ ${\href{/padicField/11.12.0.1}{12} }^{2}$ R ${\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{12}$ ${\href{/padicField/19.12.0.1}{12} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{3}$ ${\href{/padicField/31.2.0.1}{2} }^{12}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.12.0.1}{12} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{12}$ ${\href{/padicField/53.12.0.1}{12} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }^{2}$

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 Deg $24$$8$$3$$72$
\(13\) Copy content Toggle raw display 13.4.3.8a1.2$x^{12} + 9 x^{10} + 36 x^{9} + 33 x^{8} + 216 x^{7} + 495 x^{6} + 468 x^{5} + 1362 x^{4} + 2160 x^{3} + 900 x^{2} + 157 x + 8$$3$$4$$8$$C_{12}$$$[\ ]_{3}^{4}$$
13.12.1.0a1.1$x^{12} + x^{8} + 5 x^{7} + 8 x^{6} + 11 x^{5} + 3 x^{4} + x^{3} + x^{2} + 4 x + 2$$1$$12$$0$$C_{12}$$$[\ ]^{12}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*72 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*72 1.8.2t1.b.a$1$ $ 2^{3}$ \(\Q(\sqrt{-2}) \) $C_2$ (as 2T1) $1$ $-1$
*72 1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
*72 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
1.52.6t1.b.a$1$ $ 2^{2} \cdot 13 $ 6.0.1827904.1 $C_6$ (as 6T1) $0$ $-1$
1.52.6t1.b.b$1$ $ 2^{2} \cdot 13 $ 6.0.1827904.1 $C_6$ (as 6T1) $0$ $-1$
1.104.6t1.d.a$1$ $ 2^{3} \cdot 13 $ 6.0.14623232.1 $C_6$ (as 6T1) $0$ $-1$
1.13.3t1.a.a$1$ $ 13 $ 3.3.169.1 $C_3$ (as 3T1) $0$ $1$
1.13.3t1.a.b$1$ $ 13 $ 3.3.169.1 $C_3$ (as 3T1) $0$ $1$
1.104.6t1.a.a$1$ $ 2^{3} \cdot 13 $ 6.6.14623232.1 $C_6$ (as 6T1) $0$ $1$
1.104.6t1.a.b$1$ $ 2^{3} \cdot 13 $ 6.6.14623232.1 $C_6$ (as 6T1) $0$ $1$
1.104.6t1.d.b$1$ $ 2^{3} \cdot 13 $ 6.0.14623232.1 $C_6$ (as 6T1) $0$ $-1$
*72 1.16.4t1.a.a$1$ $ 2^{4}$ \(\Q(\zeta_{16})^+\) $C_4$ (as 4T1) $0$ $1$
*72 1.16.4t1.b.a$1$ $ 2^{4}$ 4.0.2048.2 $C_4$ (as 4T1) $0$ $-1$
*72 1.16.4t1.b.b$1$ $ 2^{4}$ 4.0.2048.2 $C_4$ (as 4T1) $0$ $-1$
*72 1.16.4t1.a.b$1$ $ 2^{4}$ \(\Q(\zeta_{16})^+\) $C_4$ (as 4T1) $0$ $1$
1.208.12t1.b.a$1$ $ 2^{4} \cdot 13 $ 12.12.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $1$
1.208.12t1.a.a$1$ $ 2^{4} \cdot 13 $ 12.0.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $-1$
1.208.12t1.a.b$1$ $ 2^{4} \cdot 13 $ 12.0.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $-1$
1.208.12t1.b.b$1$ $ 2^{4} \cdot 13 $ 12.12.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $1$
1.208.12t1.b.c$1$ $ 2^{4} \cdot 13 $ 12.12.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $1$
1.208.12t1.b.d$1$ $ 2^{4} \cdot 13 $ 12.12.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $1$
1.208.12t1.a.c$1$ $ 2^{4} \cdot 13 $ 12.0.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $-1$
1.208.12t1.a.d$1$ $ 2^{4} \cdot 13 $ 12.0.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $-1$
2.676.3t2.b.a$2$ $ 2^{2} \cdot 13^{2}$ 3.1.676.1 $S_3$ (as 3T2) $1$ $0$
2.10816.6t3.d.a$2$ $ 2^{6} \cdot 13^{2}$ 6.0.58492928.3 $D_{6}$ (as 6T3) $1$ $0$
*72 2.832.12t18.b.a$2$ $ 2^{6} \cdot 13 $ 12.0.479174066176.4 $C_6\times S_3$ (as 12T18) $0$ $0$
*72 2.832.12t18.b.b$2$ $ 2^{6} \cdot 13 $ 12.0.479174066176.4 $C_6\times S_3$ (as 12T18) $0$ $0$
*72 2.52.6t5.b.a$2$ $ 2^{2} \cdot 13 $ 6.0.10816.1 $S_3\times C_3$ (as 6T5) $0$ $0$
*72 2.52.6t5.b.b$2$ $ 2^{2} \cdot 13 $ 6.0.10816.1 $S_3\times C_3$ (as 6T5) $0$ $0$
2.43264.12t11.b.a$2$ $ 2^{8} \cdot 13^{2}$ 12.4.28028294152300003328.26 $S_3 \times C_4$ (as 12T11) $0$ $0$
2.43264.12t11.b.b$2$ $ 2^{8} \cdot 13^{2}$ 12.4.28028294152300003328.26 $S_3 \times C_4$ (as 12T11) $0$ $0$
*72 2.3328.24t65.a.a$2$ $ 2^{8} \cdot 13 $ 24.0.3852179415897489839182437154816.1 $S_3\times C_{12}$ (as 24T65) $0$ $0$
*72 2.3328.24t65.a.b$2$ $ 2^{8} \cdot 13 $ 24.0.3852179415897489839182437154816.1 $S_3\times C_{12}$ (as 24T65) $0$ $0$
*72 2.3328.24t65.a.c$2$ $ 2^{8} \cdot 13 $ 24.0.3852179415897489839182437154816.1 $S_3\times C_{12}$ (as 24T65) $0$ $0$
*72 2.3328.24t65.a.d$2$ $ 2^{8} \cdot 13 $ 24.0.3852179415897489839182437154816.1 $S_3\times C_{12}$ (as 24T65) $0$ $0$

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)