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Properties

Label 14.0.9176374064424843.1
Degree $14$
Signature $[0, 7]$
Discriminant $-9.176\times 10^{15}$
Root discriminant \(13.81\)
Ramified primes $3,127$
Class number $1$
Class group trivial
Galois group $C_7 \wr C_2$ (as 14T8)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 2*x^12 + 11*x^11 - 12*x^10 - 27*x^9 + 59*x^8 + 9*x^7 - 110*x^6 + 92*x^5 + 26*x^4 - 80*x^3 + 55*x^2 - 22*x + 7)
 
Copy content gp:K = bnfinit(y^14 - 4*y^13 + 2*y^12 + 11*y^11 - 12*y^10 - 27*y^9 + 59*y^8 + 9*y^7 - 110*y^6 + 92*y^5 + 26*y^4 - 80*y^3 + 55*y^2 - 22*y + 7, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 4*x^13 + 2*x^12 + 11*x^11 - 12*x^10 - 27*x^9 + 59*x^8 + 9*x^7 - 110*x^6 + 92*x^5 + 26*x^4 - 80*x^3 + 55*x^2 - 22*x + 7);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 4*x^13 + 2*x^12 + 11*x^11 - 12*x^10 - 27*x^9 + 59*x^8 + 9*x^7 - 110*x^6 + 92*x^5 + 26*x^4 - 80*x^3 + 55*x^2 - 22*x + 7)
 

\( x^{14} - 4 x^{13} + 2 x^{12} + 11 x^{11} - 12 x^{10} - 27 x^{9} + 59 x^{8} + 9 x^{7} - 110 x^{6} + \cdots + 7 \) 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:  $14$
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:  $[0, 7]$
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:   \(-9176374064424843\) \(\medspace = -\,3^{7}\cdot 127^{6}\) 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:  \(13.81\)
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:  $3^{1/2}127^{6/7}\approx 110.10852872333544$
Ramified primes:   \(3\), \(127\) 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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:   $C_7$
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.
Maximal CM subfield:  \(\Q(\sqrt{-3}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}-\frac{2}{7}a^{10}+\frac{3}{7}a^{9}+\frac{1}{7}a^{8}-\frac{1}{7}a^{6}-\frac{2}{7}a^{5}-\frac{2}{7}a^{4}+\frac{1}{7}a^{3}+\frac{3}{7}a^{2}-\frac{2}{7}a$, $\frac{1}{401509283}a^{13}+\frac{23697644}{401509283}a^{12}-\frac{161403013}{401509283}a^{11}-\frac{524844}{57358469}a^{10}-\frac{107151959}{401509283}a^{9}+\frac{92722404}{401509283}a^{8}-\frac{22514213}{401509283}a^{7}+\frac{10324904}{57358469}a^{6}+\frac{186080995}{401509283}a^{5}-\frac{181507643}{401509283}a^{4}+\frac{72660799}{401509283}a^{3}-\frac{188453280}{401509283}a^{2}-\frac{71118645}{401509283}a-\frac{23620139}{57358469}$ 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:  Trivial group, which has order $1$
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:  $6$
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:   \( \frac{203093}{1170581} a^{13} - \frac{609893}{1170581} a^{12} - \frac{235293}{1170581} a^{11} + \frac{2061252}{1170581} a^{10} - \frac{305939}{1170581} a^{9} - \frac{5821148}{1170581} a^{8} + \frac{5812218}{1170581} a^{7} + \frac{7975808}{1170581} a^{6} - \frac{14076104}{1170581} a^{5} + \frac{4265446}{1170581} a^{4} + \frac{7733399}{1170581} a^{3} - \frac{7890974}{1170581} a^{2} + \frac{5172954}{1170581} a - \frac{971267}{1170581} \)  (order $6$) 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:   $\frac{92081697}{401509283}a^{13}-\frac{279469206}{401509283}a^{12}-\frac{135027174}{401509283}a^{11}+\frac{146538268}{57358469}a^{10}-\frac{74754258}{401509283}a^{9}-\frac{2897682013}{401509283}a^{8}+\frac{2478599192}{401509283}a^{7}+\frac{637420646}{57358469}a^{6}-\frac{6771372333}{401509283}a^{5}+\frac{610773917}{401509283}a^{4}+\frac{4165751794}{401509283}a^{3}-\frac{3072153911}{401509283}a^{2}+\frac{1530276136}{401509283}a-\frac{100260948}{57358469}$, $\frac{64283140}{401509283}a^{13}-\frac{233899515}{401509283}a^{12}+\frac{43509421}{401509283}a^{11}+\frac{106391526}{57358469}a^{10}-\frac{556399260}{401509283}a^{9}-\frac{2001465056}{401509283}a^{8}+\frac{3282400074}{401509283}a^{7}+\frac{272344593}{57358469}a^{6}-\frac{7017112568}{401509283}a^{5}+\frac{3546233464}{401509283}a^{4}+\frac{3981754943}{401509283}a^{3}-\frac{4568995212}{401509283}a^{2}+\frac{1548697933}{401509283}a-\frac{2102939}{57358469}$, $\frac{76824676}{401509283}a^{13}-\frac{235869777}{401509283}a^{12}-\frac{82042124}{401509283}a^{11}+\frac{116044778}{57358469}a^{10}-\frac{112561257}{401509283}a^{9}-\frac{2391181089}{401509283}a^{8}+\frac{2274327939}{401509283}a^{7}+\frac{479909440}{57358469}a^{6}-\frac{5589872887}{401509283}a^{5}+\frac{795534800}{401509283}a^{4}+\frac{4035359165}{401509283}a^{3}-\frac{2298359992}{401509283}a^{2}+\frac{1027180048}{401509283}a+\frac{34999329}{57358469}$, $\frac{36506991}{401509283}a^{13}-\frac{81512136}{401509283}a^{12}-\frac{102557307}{401509283}a^{11}+\frac{39444376}{57358469}a^{10}+\frac{188257124}{401509283}a^{9}-\frac{905366447}{401509283}a^{8}+\frac{269791953}{401509283}a^{7}+\frac{244192374}{57358469}a^{6}-\frac{981747677}{401509283}a^{5}-\frac{450267761}{401509283}a^{4}+\frac{935517600}{401509283}a^{3}+\frac{227701555}{401509283}a^{2}+\frac{154263541}{401509283}a+\frac{5215352}{57358469}$, $\frac{52817620}{401509283}a^{13}-\frac{182009244}{401509283}a^{12}-\frac{18032158}{401509283}a^{11}+\frac{91294410}{57358469}a^{10}-\frac{243261637}{401509283}a^{9}-\frac{1805860760}{401509283}a^{8}+\frac{2093609474}{401509283}a^{7}+\frac{343419878}{57358469}a^{6}-\frac{5072853292}{401509283}a^{5}+\frac{790606064}{401509283}a^{4}+\frac{3469655233}{401509283}a^{3}-\frac{1944180506}{401509283}a^{2}+\frac{408318247}{401509283}a-\frac{34360593}{57358469}$, $\frac{80201719}{401509283}a^{13}-\frac{242993610}{401509283}a^{12}-\frac{85228863}{401509283}a^{11}+\frac{112950921}{57358469}a^{10}-\frac{85842884}{401509283}a^{9}-\frac{2242040156}{401509283}a^{8}+\frac{2212781792}{401509283}a^{7}+\frac{419415366}{57358469}a^{6}-\frac{5164833906}{401509283}a^{5}+\frac{1697377333}{401509283}a^{4}+\frac{2344595956}{401509283}a^{3}-\frac{3061279343}{401509283}a^{2}+\frac{2202536334}{401509283}a-\frac{61570108}{57358469}$ 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:  \( 620.9879838596519 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 620.9879838596519 \cdot 1}{6\cdot\sqrt{9176374064424843}}\cr\approx \mathstrut & 0.417691339283327 \end{aligned}\]

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^14 - 4*x^13 + 2*x^12 + 11*x^11 - 12*x^10 - 27*x^9 + 59*x^8 + 9*x^7 - 110*x^6 + 92*x^5 + 26*x^4 - 80*x^3 + 55*x^2 - 22*x + 7) 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^14 - 4*x^13 + 2*x^12 + 11*x^11 - 12*x^10 - 27*x^9 + 59*x^8 + 9*x^7 - 110*x^6 + 92*x^5 + 26*x^4 - 80*x^3 + 55*x^2 - 22*x + 7, 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^14 - 4*x^13 + 2*x^12 + 11*x^11 - 12*x^10 - 27*x^9 + 59*x^8 + 9*x^7 - 110*x^6 + 92*x^5 + 26*x^4 - 80*x^3 + 55*x^2 - 22*x + 7); 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^14 - 4*x^13 + 2*x^12 + 11*x^11 - 12*x^10 - 27*x^9 + 59*x^8 + 9*x^7 - 110*x^6 + 92*x^5 + 26*x^4 - 80*x^3 + 55*x^2 - 22*x + 7); 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

$C_7\times D_7$ (as 14T8):

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 98
The 35 conjugacy class representatives for $C_7 \wr C_2$
Character table for $C_7 \wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \)

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 14 siblings: deg 14, deg 14
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 ${\href{/padicField/2.14.0.1}{14} }$ R ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{7}$ ${\href{/padicField/11.14.0.1}{14} }$ ${\href{/padicField/13.7.0.1}{7} }^{2}$ ${\href{/padicField/17.14.0.1}{14} }$ ${\href{/padicField/19.7.0.1}{7} }^{2}$ ${\href{/padicField/23.14.0.1}{14} }$ ${\href{/padicField/29.14.0.1}{14} }$ ${\href{/padicField/31.7.0.1}{7} }^{2}$ ${\href{/padicField/37.7.0.1}{7} }^{2}$ ${\href{/padicField/41.14.0.1}{14} }$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.14.0.1}{14} }$ ${\href{/padicField/59.2.0.1}{2} }^{7}$

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 3.7.2.7a1.2$x^{14} + 4 x^{9} + 2 x^{7} + 4 x^{4} + 4 x^{2} + 4$$2$$7$$7$$C_{14}$$$[\ ]_{2}^{7}$$
\(127\) Copy content Toggle raw display $\Q_{127}$$x + 124$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{127}$$x + 124$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{127}$$x + 124$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{127}$$x + 124$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{127}$$x + 124$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{127}$$x + 124$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{127}$$x + 124$$1$$1$$0$Trivial$$[\ ]$$
127.1.7.6a1.1$x^{7} + 127$$7$$1$$6$$C_7$$$[\ ]_{7}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*98 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*98 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.381.14t1.a.f$1$ $ 3 \cdot 127 $ 14.0.38502899391974811462791218827.1 $C_{14}$ (as 14T1) $0$ $-1$
1.381.14t1.a.e$1$ $ 3 \cdot 127 $ 14.0.38502899391974811462791218827.1 $C_{14}$ (as 14T1) $0$ $-1$
1.381.14t1.a.c$1$ $ 3 \cdot 127 $ 14.0.38502899391974811462791218827.1 $C_{14}$ (as 14T1) $0$ $-1$
1.381.14t1.a.d$1$ $ 3 \cdot 127 $ 14.0.38502899391974811462791218827.1 $C_{14}$ (as 14T1) $0$ $-1$
1.381.14t1.a.b$1$ $ 3 \cdot 127 $ 14.0.38502899391974811462791218827.1 $C_{14}$ (as 14T1) $0$ $-1$
1.127.7t1.a.e$1$ $ 127 $ 7.7.4195872914689.1 $C_7$ (as 7T1) $0$ $1$
1.127.7t1.a.c$1$ $ 127 $ 7.7.4195872914689.1 $C_7$ (as 7T1) $0$ $1$
1.127.7t1.a.b$1$ $ 127 $ 7.7.4195872914689.1 $C_7$ (as 7T1) $0$ $1$
1.127.7t1.a.a$1$ $ 127 $ 7.7.4195872914689.1 $C_7$ (as 7T1) $0$ $1$
1.127.7t1.a.d$1$ $ 127 $ 7.7.4195872914689.1 $C_7$ (as 7T1) $0$ $1$
1.381.14t1.a.a$1$ $ 3 \cdot 127 $ 14.0.38502899391974811462791218827.1 $C_{14}$ (as 14T1) $0$ $-1$
1.127.7t1.a.f$1$ $ 127 $ 7.7.4195872914689.1 $C_7$ (as 7T1) $0$ $1$
*98 2.381.14t8.a.e$2$ $ 3 \cdot 127 $ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.48387.7t2.a.b$2$ $ 3 \cdot 127^{2}$ 7.1.113288568696603.1 $D_{7}$ (as 7T2) $1$ $0$
2.48387.14t8.a.a$2$ $ 3 \cdot 127^{2}$ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.48387.14t8.b.b$2$ $ 3 \cdot 127^{2}$ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.48387.14t8.a.e$2$ $ 3 \cdot 127^{2}$ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.381.14t8.a.d$2$ $ 3 \cdot 127 $ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.48387.14t8.b.e$2$ $ 3 \cdot 127^{2}$ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.48387.14t8.a.f$2$ $ 3 \cdot 127^{2}$ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.48387.7t2.a.a$2$ $ 3 \cdot 127^{2}$ 7.1.113288568696603.1 $D_{7}$ (as 7T2) $1$ $0$
2.48387.14t8.b.c$2$ $ 3 \cdot 127^{2}$ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.381.14t8.a.f$2$ $ 3 \cdot 127 $ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.381.14t8.a.a$2$ $ 3 \cdot 127 $ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.48387.14t8.a.d$2$ $ 3 \cdot 127^{2}$ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.48387.14t8.b.d$2$ $ 3 \cdot 127^{2}$ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.48387.14t8.a.b$2$ $ 3 \cdot 127^{2}$ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.381.14t8.a.b$2$ $ 3 \cdot 127 $ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.381.14t8.a.c$2$ $ 3 \cdot 127 $ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.48387.7t2.a.c$2$ $ 3 \cdot 127^{2}$ 7.1.113288568696603.1 $D_{7}$ (as 7T2) $1$ $0$
2.48387.14t8.b.f$2$ $ 3 \cdot 127^{2}$ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.48387.14t8.a.c$2$ $ 3 \cdot 127^{2}$ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.48387.14t8.b.a$2$ $ 3 \cdot 127^{2}$ 14.0.9176374064424843.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$

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)