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Properties

Label 14.0.5205914289462607.1
Degree $14$
Signature $[0, 7]$
Discriminant $-5.206\times 10^{15}$
Root discriminant \(13.26\)
Ramified primes $7,43$
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 - 2*x^13 + 2*x^12 - 5*x^11 + 8*x^10 + 2*x^9 - 6*x^8 + 11*x^7 - 3*x^6 + 9*x^5 + 18*x^4 - 8*x^3 + x^2 + 14*x + 7)
 
Copy content gp:K = bnfinit(y^14 - 2*y^13 + 2*y^12 - 5*y^11 + 8*y^10 + 2*y^9 - 6*y^8 + 11*y^7 - 3*y^6 + 9*y^5 + 18*y^4 - 8*y^3 + y^2 + 14*y + 7, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 2*x^13 + 2*x^12 - 5*x^11 + 8*x^10 + 2*x^9 - 6*x^8 + 11*x^7 - 3*x^6 + 9*x^5 + 18*x^4 - 8*x^3 + x^2 + 14*x + 7);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 2*x^13 + 2*x^12 - 5*x^11 + 8*x^10 + 2*x^9 - 6*x^8 + 11*x^7 - 3*x^6 + 9*x^5 + 18*x^4 - 8*x^3 + x^2 + 14*x + 7)
 

\( x^{14} - 2 x^{13} + 2 x^{12} - 5 x^{11} + 8 x^{10} + 2 x^{9} - 6 x^{8} + 11 x^{7} - 3 x^{6} + 9 x^{5} + \cdots + 7 \) 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:   \(-5205914289462607\) \(\medspace = -\,7^{7}\cdot 43^{6}\) 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:  \(13.26\)
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:  $7^{1/2}43^{6/7}\approx 66.47599386519646$
Ramified primes:   \(7\), \(43\) 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:  \(\Q(\sqrt{-7}) \)
$\Aut(K/\Q)$:   $C_7$
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:  \(\Q(\sqrt{-7}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7}a^{10}+\frac{2}{7}a^{9}+\frac{3}{7}a^{8}+\frac{1}{7}a^{7}+\frac{3}{7}a^{6}+\frac{2}{7}a^{5}-\frac{2}{7}a^{4}+\frac{2}{7}a^{3}+\frac{2}{7}a^{2}$, $\frac{1}{7}a^{11}-\frac{1}{7}a^{9}+\frac{2}{7}a^{8}+\frac{1}{7}a^{7}+\frac{3}{7}a^{6}+\frac{1}{7}a^{5}-\frac{1}{7}a^{4}-\frac{2}{7}a^{3}+\frac{3}{7}a^{2}$, $\frac{1}{7}a^{12}-\frac{3}{7}a^{9}-\frac{3}{7}a^{8}-\frac{3}{7}a^{7}-\frac{3}{7}a^{6}+\frac{1}{7}a^{5}+\frac{3}{7}a^{4}-\frac{2}{7}a^{3}+\frac{2}{7}a^{2}$, $\frac{1}{1474151}a^{13}-\frac{23202}{1474151}a^{12}+\frac{10694}{1474151}a^{11}-\frac{22251}{1474151}a^{10}+\frac{98792}{210593}a^{9}+\frac{417500}{1474151}a^{8}+\frac{435622}{1474151}a^{7}+\frac{348867}{1474151}a^{6}-\frac{59260}{210593}a^{5}+\frac{145095}{1474151}a^{4}+\frac{335716}{1474151}a^{3}-\frac{250289}{1474151}a^{2}+\frac{33515}{210593}a-\frac{38642}{210593}$ 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:   \( -1 \)  (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:   $\frac{122282}{1474151}a^{13}-\frac{288661}{1474151}a^{12}+\frac{46052}{210593}a^{11}-\frac{456408}{1474151}a^{10}+\frac{817223}{1474151}a^{9}+\frac{358754}{1474151}a^{8}-\frac{2244097}{1474151}a^{7}+\frac{468398}{210593}a^{6}-\frac{325702}{1474151}a^{5}+\frac{1520691}{1474151}a^{4}+\frac{3868131}{1474151}a^{3}-\frac{3728296}{1474151}a^{2}+\frac{352043}{210593}a+\frac{696469}{210593}$, $\frac{137623}{1474151}a^{13}-\frac{538966}{1474151}a^{12}+\frac{958850}{1474151}a^{11}-\frac{1490711}{1474151}a^{10}+\frac{2856096}{1474151}a^{9}-\frac{415449}{210593}a^{8}-\frac{8734}{1474151}a^{7}+\frac{2815745}{1474151}a^{6}-\frac{2907164}{1474151}a^{5}+\frac{629054}{210593}a^{4}-\frac{1229353}{1474151}a^{3}-\frac{2195525}{1474151}a^{2}+\frac{658738}{210593}a+\frac{77063}{210593}$, $\frac{58086}{210593}a^{13}-\frac{1097948}{1474151}a^{12}+\frac{1562268}{1474151}a^{11}-\frac{3174124}{1474151}a^{10}+\frac{5507085}{1474151}a^{9}-\frac{2892893}{1474151}a^{8}-\frac{51830}{210593}a^{7}+\frac{4473005}{1474151}a^{6}-\frac{5235056}{1474151}a^{5}+\frac{7554332}{1474151}a^{4}+\frac{1889850}{1474151}a^{3}-\frac{5469111}{1474151}a^{2}+\frac{424779}{210593}a+\frac{429246}{210593}$, $\frac{43894}{1474151}a^{13}-\frac{120}{210593}a^{12}-\frac{9361}{1474151}a^{11}-\frac{165653}{1474151}a^{10}-\frac{32091}{1474151}a^{9}+\frac{363326}{1474151}a^{8}+\frac{87318}{210593}a^{7}+\frac{319289}{1474151}a^{6}+\frac{1025258}{1474151}a^{5}-\frac{795948}{1474151}a^{4}+\frac{1778859}{1474151}a^{3}+\frac{1083223}{1474151}a^{2}+\frac{115305}{210593}a+\frac{174667}{210593}$, $\frac{214710}{1474151}a^{13}-\frac{545191}{1474151}a^{12}+\frac{855633}{1474151}a^{11}-\frac{1684156}{1474151}a^{10}+\frac{2606997}{1474151}a^{9}-\frac{1486717}{1474151}a^{8}+\frac{45786}{1474151}a^{7}+\frac{3831853}{1474151}a^{6}-\frac{3063605}{1474151}a^{5}+\frac{557863}{210593}a^{4}+\frac{652692}{1474151}a^{3}-\frac{1693008}{1474151}a^{2}+\frac{464026}{210593}a+\frac{329787}{210593}$, $\frac{96617}{1474151}a^{13}-\frac{155742}{1474151}a^{12}+\frac{263533}{1474151}a^{11}-\frac{512709}{1474151}a^{10}+\frac{69532}{210593}a^{9}+\frac{193094}{1474151}a^{8}-\frac{89458}{210593}a^{7}+\frac{3179219}{1474151}a^{6}-\frac{1131296}{1474151}a^{5}+\frac{3047686}{1474151}a^{4}+\frac{2023656}{1474151}a^{3}-\frac{199309}{1474151}a^{2}+\frac{461973}{210593}a+\frac{129183}{210593}$ 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:  \( 154.6496312968527 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 154.6496312968527 \cdot 1}{2\cdot\sqrt{5205914289462607}}\cr\approx \mathstrut & 0.414313862894185 \end{aligned}\]

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 - 2*x^13 + 2*x^12 - 5*x^11 + 8*x^10 + 2*x^9 - 6*x^8 + 11*x^7 - 3*x^6 + 9*x^5 + 18*x^4 - 8*x^3 + x^2 + 14*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 - 2*x^13 + 2*x^12 - 5*x^11 + 8*x^10 + 2*x^9 - 6*x^8 + 11*x^7 - 3*x^6 + 9*x^5 + 18*x^4 - 8*x^3 + x^2 + 14*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 - 2*x^13 + 2*x^12 - 5*x^11 + 8*x^10 + 2*x^9 - 6*x^8 + 11*x^7 - 3*x^6 + 9*x^5 + 18*x^4 - 8*x^3 + x^2 + 14*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 - 2*x^13 + 2*x^12 - 5*x^11 + 8*x^10 + 2*x^9 - 6*x^8 + 11*x^7 - 3*x^6 + 9*x^5 + 18*x^4 - 8*x^3 + x^2 + 14*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

\(\Q(\sqrt{-7}) \)

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: 14.0.32908474225670013957008743.2, 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 ${\href{/padicField/2.7.0.1}{7} }^{2}$ ${\href{/padicField/3.14.0.1}{14} }$ ${\href{/padicField/5.14.0.1}{14} }$ R ${\href{/padicField/11.7.0.1}{7} }^{2}$ ${\href{/padicField/13.14.0.1}{14} }$ ${\href{/padicField/17.14.0.1}{14} }$ ${\href{/padicField/19.14.0.1}{14} }$ ${\href{/padicField/23.7.0.1}{7} }^{2}$ ${\href{/padicField/29.7.0.1}{7} }^{2}$ ${\href{/padicField/31.14.0.1}{14} }$ ${\href{/padicField/37.7.0.1}{7} }^{2}$ ${\href{/padicField/41.14.0.1}{14} }$ R ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{7}$ ${\href{/padicField/59.14.0.1}{14} }$

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
\(7\) Copy content Toggle raw display 7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(43\) Copy content Toggle raw display 43.7.1.0a1.1$x^{7} + 42 x^{2} + 7 x + 40$$1$$7$$0$$C_7$$$[\ ]^{7}$$
43.1.7.6a1.2$x^{7} + 129$$7$$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 1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
1.301.14t1.a.a$1$ $ 7 \cdot 43 $ 14.0.32908474225670013957008743.1 $C_{14}$ (as 14T1) $0$ $-1$
1.43.7t1.a.b$1$ $ 43 $ 7.7.6321363049.1 $C_7$ (as 7T1) $0$ $1$
1.301.14t1.a.c$1$ $ 7 \cdot 43 $ 14.0.32908474225670013957008743.1 $C_{14}$ (as 14T1) $0$ $-1$
1.301.14t1.a.d$1$ $ 7 \cdot 43 $ 14.0.32908474225670013957008743.1 $C_{14}$ (as 14T1) $0$ $-1$
1.301.14t1.a.f$1$ $ 7 \cdot 43 $ 14.0.32908474225670013957008743.1 $C_{14}$ (as 14T1) $0$ $-1$
1.43.7t1.a.d$1$ $ 43 $ 7.7.6321363049.1 $C_7$ (as 7T1) $0$ $1$
1.301.14t1.a.b$1$ $ 7 \cdot 43 $ 14.0.32908474225670013957008743.1 $C_{14}$ (as 14T1) $0$ $-1$
1.301.14t1.a.e$1$ $ 7 \cdot 43 $ 14.0.32908474225670013957008743.1 $C_{14}$ (as 14T1) $0$ $-1$
1.43.7t1.a.f$1$ $ 43 $ 7.7.6321363049.1 $C_7$ (as 7T1) $0$ $1$
1.43.7t1.a.c$1$ $ 43 $ 7.7.6321363049.1 $C_7$ (as 7T1) $0$ $1$
1.43.7t1.a.a$1$ $ 43 $ 7.7.6321363049.1 $C_7$ (as 7T1) $0$ $1$
1.43.7t1.a.e$1$ $ 43 $ 7.7.6321363049.1 $C_7$ (as 7T1) $0$ $1$
2.12943.7t2.a.b$2$ $ 7 \cdot 43^{2}$ 7.1.2168227525807.1 $D_{7}$ (as 7T2) $1$ $0$
*98 2.301.14t8.b.c$2$ $ 7 \cdot 43 $ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.301.14t8.b.a$2$ $ 7 \cdot 43 $ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.12943.14t8.c.e$2$ $ 7 \cdot 43^{2}$ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.12943.14t8.c.f$2$ $ 7 \cdot 43^{2}$ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.12943.14t8.d.a$2$ $ 7 \cdot 43^{2}$ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.301.14t8.b.e$2$ $ 7 \cdot 43 $ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.301.14t8.b.d$2$ $ 7 \cdot 43 $ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.12943.7t2.a.c$2$ $ 7 \cdot 43^{2}$ 7.1.2168227525807.1 $D_{7}$ (as 7T2) $1$ $0$
2.12943.14t8.d.c$2$ $ 7 \cdot 43^{2}$ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.301.14t8.b.b$2$ $ 7 \cdot 43 $ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.12943.14t8.d.f$2$ $ 7 \cdot 43^{2}$ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.301.14t8.b.f$2$ $ 7 \cdot 43 $ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.12943.14t8.d.e$2$ $ 7 \cdot 43^{2}$ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.12943.14t8.c.a$2$ $ 7 \cdot 43^{2}$ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.12943.14t8.c.c$2$ $ 7 \cdot 43^{2}$ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.12943.14t8.c.b$2$ $ 7 \cdot 43^{2}$ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.12943.7t2.a.a$2$ $ 7 \cdot 43^{2}$ 7.1.2168227525807.1 $D_{7}$ (as 7T2) $1$ $0$
2.12943.14t8.c.d$2$ $ 7 \cdot 43^{2}$ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.12943.14t8.d.b$2$ $ 7 \cdot 43^{2}$ 14.0.5205914289462607.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.12943.14t8.d.d$2$ $ 7 \cdot 43^{2}$ 14.0.5205914289462607.1 $C_7 \wr C_2$ (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)