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Properties

Label 14.0.192993869976848307.1
Degree $14$
Signature $[0, 7]$
Discriminant $-1.930\times 10^{17}$
Root discriminant \(17.17\)
Ramified primes $3,211$
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 - 6*x^13 + 19*x^12 - 36*x^11 + 34*x^10 + 7*x^9 - 24*x^8 - 225*x^7 + 1029*x^6 - 2242*x^5 + 3040*x^4 - 2658*x^3 + 1454*x^2 - 459*x + 67)
 
Copy content gp:K = bnfinit(y^14 - 6*y^13 + 19*y^12 - 36*y^11 + 34*y^10 + 7*y^9 - 24*y^8 - 225*y^7 + 1029*y^6 - 2242*y^5 + 3040*y^4 - 2658*y^3 + 1454*y^2 - 459*y + 67, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 6*x^13 + 19*x^12 - 36*x^11 + 34*x^10 + 7*x^9 - 24*x^8 - 225*x^7 + 1029*x^6 - 2242*x^5 + 3040*x^4 - 2658*x^3 + 1454*x^2 - 459*x + 67);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 6*x^13 + 19*x^12 - 36*x^11 + 34*x^10 + 7*x^9 - 24*x^8 - 225*x^7 + 1029*x^6 - 2242*x^5 + 3040*x^4 - 2658*x^3 + 1454*x^2 - 459*x + 67)
 

\( x^{14} - 6 x^{13} + 19 x^{12} - 36 x^{11} + 34 x^{10} + 7 x^{9} - 24 x^{8} - 225 x^{7} + 1029 x^{6} + \cdots + 67 \) 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:   \(-192993869976848307\) \(\medspace = -\,3^{7}\cdot 211^{6}\) 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:  \(17.17\)
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:  $3^{1/2}211^{6/7}\approx 170.1385518318596$
Ramified primes:   \(3\), \(211\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{1444405967833}a^{13}+\frac{621688225785}{1444405967833}a^{12}+\frac{94384295695}{1444405967833}a^{11}+\frac{658351238098}{1444405967833}a^{10}+\frac{467387330798}{1444405967833}a^{9}-\frac{81406258017}{1444405967833}a^{8}+\frac{532831883391}{1444405967833}a^{7}+\frac{592862709555}{1444405967833}a^{6}-\frac{37105749872}{1444405967833}a^{5}-\frac{654558370205}{1444405967833}a^{4}+\frac{577293998834}{1444405967833}a^{3}+\frac{446473878815}{1444405967833}a^{2}-\frac{710686450759}{1444405967833}a+\frac{401419802581}{1444405967833}$ 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:   \( -\frac{38733090}{192100807} a^{13} + \frac{190683410}{192100807} a^{12} - \frac{525479504}{192100807} a^{11} + \frac{805385401}{192100807} a^{10} - \frac{389326545}{192100807} a^{9} - \frac{777039560}{192100807} a^{8} + \frac{118273147}{192100807} a^{7} + \frac{8941378942}{192100807} a^{6} - \frac{30207258420}{192100807} a^{5} + \frac{53169097819}{192100807} a^{4} - \frac{56945187736}{192100807} a^{3} + \frac{35796523729}{192100807} a^{2} - \frac{11971036180}{192100807} a + \frac{1840720327}{192100807} \)  (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:   $\frac{2982040}{192100807}a^{13}-\frac{19765796}{192100807}a^{12}+\frac{63526335}{192100807}a^{11}-\frac{122213114}{192100807}a^{10}+\frac{116581385}{192100807}a^{9}+\frac{34281453}{192100807}a^{8}-\frac{108160545}{192100807}a^{7}-\frac{707712248}{192100807}a^{6}+\frac{3463231541}{192100807}a^{5}-\frac{7634308045}{192100807}a^{4}+\frac{10210841755}{192100807}a^{3}-\frac{8550352951}{192100807}a^{2}+\frac{3966731803}{192100807}a-\frac{754396703}{192100807}$, $\frac{176754946486}{1444405967833}a^{13}-\frac{782425802875}{1444405967833}a^{12}+\frac{2053868264612}{1444405967833}a^{11}-\frac{2851186089368}{1444405967833}a^{10}+\frac{841777854863}{1444405967833}a^{9}+\frac{3374016482668}{1444405967833}a^{8}+\frac{1108910512992}{1444405967833}a^{7}-\frac{39198725349498}{1444405967833}a^{6}+\frac{119015155816555}{1444405967833}a^{5}-\frac{193889262487725}{1444405967833}a^{4}+\frac{192424124865596}{1444405967833}a^{3}-\frac{109119967740478}{1444405967833}a^{2}+\frac{33233383417700}{1444405967833}a-\frac{3825532252732}{1444405967833}$, $\frac{156993000295}{1444405967833}a^{13}-\frac{668956858524}{1444405967833}a^{12}+\frac{1692950292456}{1444405967833}a^{11}-\frac{2182357535223}{1444405967833}a^{10}+\frac{233780302857}{1444405967833}a^{9}+\frac{3168924382569}{1444405967833}a^{8}+\frac{1622508433720}{1444405967833}a^{7}-\frac{34921718467916}{1444405967833}a^{6}+\frac{99235411955837}{1444405967833}a^{5}-\frac{151878639270918}{1444405967833}a^{4}+\frac{136562633632274}{1444405967833}a^{3}-\frac{63479001724634}{1444405967833}a^{2}+\frac{12678220636743}{1444405967833}a-\frac{106659375222}{1444405967833}$, $\frac{9200216549}{1444405967833}a^{13}-\frac{152278697048}{1444405967833}a^{12}+\frac{578439207063}{1444405967833}a^{11}-\frac{1321189408243}{1444405967833}a^{10}+\frac{1508938296315}{1444405967833}a^{9}+\frac{192119306247}{1444405967833}a^{8}-\frac{2282807863336}{1444405967833}a^{7}-\frac{3347583318343}{1444405967833}a^{6}+\frac{31245382177284}{1444405967833}a^{5}-\frac{79501455283027}{1444405967833}a^{4}+\frac{112290015532739}{1444405967833}a^{3}-\frac{91738990182259}{1444405967833}a^{2}+\frac{35232271659422}{1444405967833}a-\frac{4193237857481}{1444405967833}$, $\frac{609337565129}{1444405967833}a^{13}-\frac{2982992868587}{1444405967833}a^{12}+\frac{8253954539077}{1444405967833}a^{11}-\frac{12716329816661}{1444405967833}a^{10}+\frac{6432804346062}{1444405967833}a^{9}+\frac{11613553917427}{1444405967833}a^{8}-\frac{1716250630286}{1444405967833}a^{7}-\frac{139445569287568}{1444405967833}a^{6}+\frac{472343143427220}{1444405967833}a^{5}-\frac{838690661691790}{1444405967833}a^{4}+\frac{911975175457580}{1444405967833}a^{3}-\frac{593134903864795}{1444405967833}a^{2}+\frac{215520427942309}{1444405967833}a-\frac{35357108232296}{1444405967833}$, $\frac{359742798100}{1444405967833}a^{13}-\frac{1758141604781}{1444405967833}a^{12}+\frac{4898148341986}{1444405967833}a^{11}-\frac{7565425470921}{1444405967833}a^{10}+\frac{3948562158791}{1444405967833}a^{9}+\frac{6797240820314}{1444405967833}a^{8}-\frac{1236071923700}{1444405967833}a^{7}-\frac{81989412273952}{1444405967833}a^{6}+\frac{279431949605202}{1444405967833}a^{5}-\frac{499536929758603}{1444405967833}a^{4}+\frac{545976744825027}{1444405967833}a^{3}-\frac{357654714771780}{1444405967833}a^{2}+\frac{128179884255939}{1444405967833}a-\frac{19947207555766}{1444405967833}$ 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:  \( 1350.6941155064887 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 1350.6941155064887 \cdot 1}{6\cdot\sqrt{192993869976848307}}\cr\approx \mathstrut & 0.198103924217724 \end{aligned}\]

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 - 6*x^13 + 19*x^12 - 36*x^11 + 34*x^10 + 7*x^9 - 24*x^8 - 225*x^7 + 1029*x^6 - 2242*x^5 + 3040*x^4 - 2658*x^3 + 1454*x^2 - 459*x + 67) 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 - 6*x^13 + 19*x^12 - 36*x^11 + 34*x^10 + 7*x^9 - 24*x^8 - 225*x^7 + 1029*x^6 - 2242*x^5 + 3040*x^4 - 2658*x^3 + 1454*x^2 - 459*x + 67, 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 - 6*x^13 + 19*x^12 - 36*x^11 + 34*x^10 + 7*x^9 - 24*x^8 - 225*x^7 + 1029*x^6 - 2242*x^5 + 3040*x^4 - 2658*x^3 + 1454*x^2 - 459*x + 67); 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 - 6*x^13 + 19*x^12 - 36*x^11 + 34*x^10 + 7*x^9 - 24*x^8 - 225*x^7 + 1029*x^6 - 2242*x^5 + 3040*x^4 - 2658*x^3 + 1454*x^2 - 459*x + 67); 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 ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.7.0.1}{7} }^{2}$ ${\href{/padicField/11.14.0.1}{14} }$ ${\href{/padicField/13.7.0.1}{7} }^{2}$ ${\href{/padicField/17.14.0.1}{14} }$ ${\href{/padicField/19.7.0.1}{7} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{7}$ ${\href{/padicField/29.14.0.1}{14} }$ ${\href{/padicField/31.7.0.1}{7} }^{2}$ ${\href{/padicField/37.7.0.1}{7} }^{2}$ ${\href{/padicField/41.14.0.1}{14} }$ ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{7}$ ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.14.0.1}{14} }$ ${\href{/padicField/59.14.0.1}{14} }$

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}$$$[\ ]_{2}^{7}$$
\(211\) Copy content Toggle raw display Deg $7$$1$$7$$0$$C_7$$$[\ ]^{7}$$
Deg $7$$7$$1$$6$

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) $1$ $-1$
1.211.7t1.a.c$1$ $ 211 $ 7.7.88245939632761.1 $C_7$ (as 7T1) $0$ $1$
1.211.7t1.a.e$1$ $ 211 $ 7.7.88245939632761.1 $C_7$ (as 7T1) $0$ $1$
1.211.7t1.a.d$1$ $ 211 $ 7.7.88245939632761.1 $C_7$ (as 7T1) $0$ $1$
1.633.14t1.a.e$1$ $ 3 \cdot 211 $ 14.0.17030925399469881272195784585627.1 $C_{14}$ (as 14T1) $0$ $-1$
1.633.14t1.a.a$1$ $ 3 \cdot 211 $ 14.0.17030925399469881272195784585627.1 $C_{14}$ (as 14T1) $0$ $-1$
1.211.7t1.a.b$1$ $ 211 $ 7.7.88245939632761.1 $C_7$ (as 7T1) $0$ $1$
1.211.7t1.a.f$1$ $ 211 $ 7.7.88245939632761.1 $C_7$ (as 7T1) $0$ $1$
1.633.14t1.a.f$1$ $ 3 \cdot 211 $ 14.0.17030925399469881272195784585627.1 $C_{14}$ (as 14T1) $0$ $-1$
1.633.14t1.a.d$1$ $ 3 \cdot 211 $ 14.0.17030925399469881272195784585627.1 $C_{14}$ (as 14T1) $0$ $-1$
1.633.14t1.a.c$1$ $ 3 \cdot 211 $ 14.0.17030925399469881272195784585627.1 $C_{14}$ (as 14T1) $0$ $-1$
1.633.14t1.a.b$1$ $ 3 \cdot 211 $ 14.0.17030925399469881272195784585627.1 $C_{14}$ (as 14T1) $0$ $-1$
1.211.7t1.a.a$1$ $ 211 $ 7.7.88245939632761.1 $C_7$ (as 7T1) $0$ $1$
*98 2.633.14t8.a.f$2$ $ 3 \cdot 211 $ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.14t8.a.a$2$ $ 3 \cdot 211^{2}$ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.14t8.a.f$2$ $ 3 \cdot 211^{2}$ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.7t2.a.b$2$ $ 3 \cdot 211^{2}$ 7.1.2382640370084547.1 $D_{7}$ (as 7T2) $1$ $0$
2.133563.14t8.a.b$2$ $ 3 \cdot 211^{2}$ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.633.14t8.a.c$2$ $ 3 \cdot 211 $ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.14t8.b.e$2$ $ 3 \cdot 211^{2}$ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.14t8.b.a$2$ $ 3 \cdot 211^{2}$ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.14t8.a.d$2$ $ 3 \cdot 211^{2}$ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.7t2.a.c$2$ $ 3 \cdot 211^{2}$ 7.1.2382640370084547.1 $D_{7}$ (as 7T2) $1$ $0$
*98 2.633.14t8.a.b$2$ $ 3 \cdot 211 $ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.14t8.b.b$2$ $ 3 \cdot 211^{2}$ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.14t8.b.c$2$ $ 3 \cdot 211^{2}$ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.14t8.b.d$2$ $ 3 \cdot 211^{2}$ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.7t2.a.a$2$ $ 3 \cdot 211^{2}$ 7.1.2382640370084547.1 $D_{7}$ (as 7T2) $1$ $0$
*98 2.633.14t8.a.e$2$ $ 3 \cdot 211 $ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.633.14t8.a.a$2$ $ 3 \cdot 211 $ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.633.14t8.a.d$2$ $ 3 \cdot 211 $ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.14t8.a.c$2$ $ 3 \cdot 211^{2}$ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.14t8.b.f$2$ $ 3 \cdot 211^{2}$ 14.0.192993869976848307.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.133563.14t8.a.e$2$ $ 3 \cdot 211^{2}$ 14.0.192993869976848307.1 $C_7 \wr C_2$ (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)