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Properties

Label 14.0.418...987.1
Degree $14$
Signature $(0, 7)$
Discriminant $-4.190\times 10^{23}$
Root discriminant \(48.67\)
Ramified primes $3,7$
Class number $203$ (GRH)
Class group [203] (GRH)
Galois group $C_{14}$ (as 14T1)

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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 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409)
 
Copy content gp:K = bnfinit(y^14 + 21*y^12 - 42*y^11 + 350*y^10 - 553*y^9 + 2184*y^8 - 2696*y^7 + 8869*y^6 - 8701*y^5 + 18151*y^4 - 8246*y^3 + 17920*y^2 - 8148*y + 9409, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409)
 

\( x^{14} + 21 x^{12} - 42 x^{11} + 350 x^{10} - 553 x^{9} + 2184 x^{8} - 2696 x^{7} + 8869 x^{6} + \cdots + 9409 \) 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:   \(-418988153029298748294987\) \(\medspace = -\,3^{7}\cdot 7^{24}\) 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:  \(48.67\)
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:  $3^{1/2}7^{12/7}\approx 48.67434595281677$
Ramified primes:   \(3\), \(7\) 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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{14}$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(147=3\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{147}(64,·)$, $\chi_{147}(1,·)$, $\chi_{147}(134,·)$, $\chi_{147}(71,·)$, $\chi_{147}(8,·)$, $\chi_{147}(106,·)$, $\chi_{147}(43,·)$, $\chi_{147}(113,·)$, $\chi_{147}(50,·)$, $\chi_{147}(85,·)$, $\chi_{147}(22,·)$, $\chi_{147}(92,·)$, $\chi_{147}(29,·)$, $\chi_{147}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{64}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{589}a^{10}-\frac{268}{589}a^{9}-\frac{125}{589}a^{8}+\frac{265}{589}a^{7}-\frac{156}{589}a^{6}+\frac{23}{589}a^{5}-\frac{273}{589}a^{4}-\frac{198}{589}a^{3}+\frac{159}{589}a^{2}+\frac{26}{589}a-\frac{233}{589}$, $\frac{1}{589}a^{11}-\frac{91}{589}a^{9}-\frac{251}{589}a^{8}+\frac{184}{589}a^{7}+\frac{34}{589}a^{6}+\frac{1}{589}a^{5}+\frac{263}{589}a^{4}+\frac{105}{589}a^{3}+\frac{230}{589}a^{2}+\frac{256}{589}a-\frac{10}{589}$, $\frac{1}{589}a^{12}+\frac{99}{589}a^{9}-\frac{59}{589}a^{6}-\frac{118}{589}a^{3}+\frac{1}{589}$, $\frac{1}{18\cdots 53}a^{13}+\frac{5828455076247}{19\cdots 49}a^{12}-\frac{91476252389347}{18\cdots 53}a^{11}-\frac{10\cdots 66}{18\cdots 53}a^{10}+\frac{77\cdots 06}{18\cdots 53}a^{9}+\frac{17\cdots 44}{18\cdots 53}a^{8}+\frac{97\cdots 27}{18\cdots 53}a^{7}-\frac{95\cdots 11}{18\cdots 53}a^{6}+\frac{29\cdots 21}{18\cdots 53}a^{5}+\frac{54\cdots 43}{18\cdots 53}a^{4}+\frac{61\cdots 45}{18\cdots 53}a^{3}-\frac{27\cdots 95}{18\cdots 53}a^{2}+\frac{49\cdots 48}{18\cdots 53}a-\frac{43\cdots 41}{19\cdots 49}$ 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:  $C_{203}$, which has order $203$ (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:  $C_{203}$, which has order $203$ (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);
 
Relative class number:   $203$ (assuming GRH)

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:   \( -\frac{161289742466524}{1895673790255648453} a^{13} - \frac{957311663154}{19543028765522149} a^{12} - \frac{3252983796414282}{1895673790255648453} a^{11} + \frac{4785500909859803}{1895673790255648453} a^{10} - \frac{50010109733421474}{1895673790255648453} a^{9} + \frac{50386722190134414}{1895673790255648453} a^{8} - \frac{257564795016183278}{1895673790255648453} a^{7} + \frac{157348402783570740}{1895673790255648453} a^{6} - \frac{958061773957764798}{1895673790255648453} a^{5} + \frac{288377486595742334}{1895673790255648453} a^{4} - \frac{1285265248884962772}{1895673790255648453} a^{3} - \frac{1523096170019655210}{1895673790255648453} a^{2} - \frac{1081764845716455943}{1895673790255648453} a + \frac{2712612543346663}{19543028765522149} \)  (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:   $\frac{500894425233379}{18\cdots 53}a^{13}+\frac{9083208803277}{19\cdots 49}a^{12}+\frac{10\cdots 04}{18\cdots 53}a^{11}-\frac{48\cdots 58}{18\cdots 53}a^{10}+\frac{13\cdots 29}{18\cdots 53}a^{9}-\frac{30\cdots 04}{18\cdots 53}a^{8}+\frac{69\cdots 92}{18\cdots 53}a^{7}-\frac{14\cdots 59}{18\cdots 53}a^{6}+\frac{24\cdots 59}{18\cdots 53}a^{5}+\frac{14\cdots 04}{18\cdots 53}a^{4}+\frac{31\cdots 67}{18\cdots 53}a^{3}+\frac{41\cdots 43}{18\cdots 53}a^{2}+\frac{24\cdots 42}{18\cdots 53}a-\frac{16\cdots 09}{19\cdots 49}$, $\frac{1182799648905}{19\cdots 49}a^{13}+\frac{8568865761786}{19\cdots 49}a^{12}+\frac{7684286930910}{19\cdots 49}a^{11}+\frac{77847407861811}{19\cdots 49}a^{10}-\frac{326021747139292}{19\cdots 49}a^{9}+\frac{21\cdots 20}{19\cdots 49}a^{8}-\frac{59\cdots 51}{19\cdots 49}a^{7}+\frac{10\cdots 52}{19\cdots 49}a^{6}-\frac{27\cdots 81}{19\cdots 49}a^{5}+\frac{43\cdots 62}{19\cdots 49}a^{4}-\frac{86\cdots 40}{19\cdots 49}a^{3}+\frac{47\cdots 64}{19\cdots 49}a^{2}-\frac{27\cdots 47}{19\cdots 49}a-\frac{14\cdots 79}{19\cdots 49}$, $\frac{339974479466749}{18\cdots 53}a^{13}+\frac{5231873784525}{19\cdots 49}a^{12}+\frac{64\cdots 91}{18\cdots 53}a^{11}-\frac{46\cdots 05}{18\cdots 53}a^{10}+\frac{87\cdots 95}{18\cdots 53}a^{9}+\frac{19\cdots 79}{18\cdots 53}a^{8}+\frac{35\cdots 75}{18\cdots 53}a^{7}+\frac{17\cdots 34}{18\cdots 53}a^{6}+\frac{16\cdots 17}{18\cdots 53}a^{5}+\frac{10\cdots 11}{18\cdots 53}a^{4}+\frac{24\cdots 84}{18\cdots 53}a^{3}+\frac{47\cdots 73}{18\cdots 53}a^{2}+\frac{49\cdots 29}{18\cdots 53}a+\frac{45\cdots 05}{19\cdots 49}$, $\frac{8195604655819}{19\cdots 49}a^{13}+\frac{7724519215672}{19\cdots 49}a^{12}+\frac{143285362101380}{19\cdots 49}a^{11}-\frac{276452985460239}{19\cdots 49}a^{10}+\frac{18\cdots 69}{19\cdots 49}a^{9}-\frac{23\cdots 44}{19\cdots 49}a^{8}+\frac{70\cdots 03}{19\cdots 49}a^{7}-\frac{13\cdots 61}{19\cdots 49}a^{6}+\frac{22\cdots 81}{19\cdots 49}a^{5}-\frac{27\cdots 18}{19\cdots 49}a^{4}+\frac{31\cdots 55}{19\cdots 49}a^{3}-\frac{30\cdots 10}{19\cdots 49}a^{2}+\frac{18\cdots 25}{19\cdots 49}a-\frac{28\cdots 54}{19\cdots 49}$, $\frac{18709735634464}{19\cdots 49}a^{13}+\frac{5140286677554}{19\cdots 49}a^{12}+\frac{344302418627400}{19\cdots 49}a^{11}-\frac{786930198666752}{19\cdots 49}a^{10}+\frac{52\cdots 31}{19\cdots 49}a^{9}-\frac{84\cdots 64}{19\cdots 49}a^{8}+\frac{25\cdots 96}{19\cdots 49}a^{7}-\frac{42\cdots 12}{19\cdots 49}a^{6}+\frac{92\cdots 84}{19\cdots 49}a^{5}-\frac{12\cdots 64}{19\cdots 49}a^{4}+\frac{98\cdots 23}{19\cdots 49}a^{3}-\frac{13\cdots 32}{19\cdots 49}a^{2}+\frac{82\cdots 56}{19\cdots 49}a-\frac{10\cdots 06}{19\cdots 49}$, $\frac{957311663154}{19\cdots 49}a^{13}-\frac{1382482426626}{19\cdots 49}a^{12}+\frac{20501734780765}{19\cdots 49}a^{11}-\frac{66405155977958}{19\cdots 49}a^{10}+\frac{400067065916014}{19\cdots 49}a^{9}-\frac{976206211656754}{19\cdots 49}a^{8}+\frac{28\cdots 12}{19\cdots 49}a^{7}-\frac{48\cdots 14}{19\cdots 49}a^{6}+\frac{11\cdots 70}{19\cdots 49}a^{5}-\frac{16\cdots 16}{19\cdots 49}a^{4}+\frac{29\cdots 62}{19\cdots 49}a^{3}-\frac{18\cdots 21}{19\cdots 49}a^{2}+\frac{10\cdots 53}{19\cdots 49}a+\frac{23\cdots 70}{19\cdots 49}$ 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:  \( 35256.6897369 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 35256.6897369 \cdot 203}{6\cdot\sqrt{418988153029298748294987}}\cr\approx \mathstrut & 0.712433820745 \end{aligned}\] (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^14 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409) 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 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409, 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 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409); 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 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409); 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

$C_{14}$ (as 14T1):

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 cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 7.7.13841287201.1

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(L)]
 
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 ${\href{/padicField/2.14.0.1}{14} }$ R ${\href{/padicField/5.14.0.1}{14} }$ R ${\href{/padicField/11.14.0.1}{14} }$ ${\href{/padicField/13.7.0.1}{7} }^{2}$ ${\href{/padicField/17.14.0.1}{14} }$ ${\href{/padicField/19.1.0.1}{1} }^{14}$ ${\href{/padicField/23.14.0.1}{14} }$ ${\href{/padicField/29.14.0.1}{14} }$ ${\href{/padicField/31.1.0.1}{1} }^{14}$ ${\href{/padicField/37.7.0.1}{7} }^{2}$ ${\href{/padicField/41.14.0.1}{14} }$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.14.0.1}{14} }$ ${\href{/padicField/59.14.0.1}{14} }$

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}$$$[\ ]_{2}^{7}$$
\(7\) Copy content Toggle raw display 7.1.7.12a6.1$x^{7} + 42 x^{6} + 7$$7$$1$$12$$C_7$$$[2]$$
7.1.7.12a6.1$x^{7} + 42 x^{6} + 7$$7$$1$$12$$C_7$$$[2]$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*14 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*14 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
*14 1.49.7t1.a.a$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
*14 1.147.14t1.a.d$1$ $ 3 \cdot 7^{2}$ 14.0.418988153029298748294987.1 $C_{14}$ (as 14T1) $0$ $-1$
*14 1.49.7t1.a.b$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
*14 1.147.14t1.a.f$1$ $ 3 \cdot 7^{2}$ 14.0.418988153029298748294987.1 $C_{14}$ (as 14T1) $0$ $-1$
*14 1.49.7t1.a.c$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
*14 1.147.14t1.a.b$1$ $ 3 \cdot 7^{2}$ 14.0.418988153029298748294987.1 $C_{14}$ (as 14T1) $0$ $-1$
*14 1.49.7t1.a.d$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
*14 1.147.14t1.a.c$1$ $ 3 \cdot 7^{2}$ 14.0.418988153029298748294987.1 $C_{14}$ (as 14T1) $0$ $-1$
*14 1.49.7t1.a.e$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
*14 1.147.14t1.a.e$1$ $ 3 \cdot 7^{2}$ 14.0.418988153029298748294987.1 $C_{14}$ (as 14T1) $0$ $-1$
*14 1.49.7t1.a.f$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
*14 1.147.14t1.a.a$1$ $ 3 \cdot 7^{2}$ 14.0.418988153029298748294987.1 $C_{14}$ (as 14T1) $0$ $-1$

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)