Normalized defining polynomial
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)
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)
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);
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)
\( x^{14} + 21 x^{12} - 42 x^{11} + 350 x^{10} - 553 x^{9} + 2184 x^{8} - 2696 x^{7} + 8869 x^{6} + \cdots + 9409 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-418988153029298748294987\)
\(\medspace = -\,3^{7}\cdot 7^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(48.67\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $3^{1/2}7^{12/7}\approx 48.67434595281677$ | ||
| Ramified primes: |
\(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $14$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{589}a^{10}-\frac{268}{589}a^{9}-\frac{125}{589}a^{8}+\frac{265}{589}a^{7}-\frac{156}{589}a^{6}+\frac{23}{589}a^{5}-\frac{273}{589}a^{4}-\frac{198}{589}a^{3}+\frac{159}{589}a^{2}+\frac{26}{589}a-\frac{233}{589}$, $\frac{1}{589}a^{11}-\frac{91}{589}a^{9}-\frac{251}{589}a^{8}+\frac{184}{589}a^{7}+\frac{34}{589}a^{6}+\frac{1}{589}a^{5}+\frac{263}{589}a^{4}+\frac{105}{589}a^{3}+\frac{230}{589}a^{2}+\frac{256}{589}a-\frac{10}{589}$, $\frac{1}{589}a^{12}+\frac{99}{589}a^{9}-\frac{59}{589}a^{6}-\frac{118}{589}a^{3}+\frac{1}{589}$, $\frac{1}{18\!\cdots\!53}a^{13}+\frac{5828455076247}{19\!\cdots\!49}a^{12}-\frac{91476252389347}{18\!\cdots\!53}a^{11}-\frac{10\!\cdots\!66}{18\!\cdots\!53}a^{10}+\frac{77\!\cdots\!06}{18\!\cdots\!53}a^{9}+\frac{17\!\cdots\!44}{18\!\cdots\!53}a^{8}+\frac{97\!\cdots\!27}{18\!\cdots\!53}a^{7}-\frac{95\!\cdots\!11}{18\!\cdots\!53}a^{6}+\frac{29\!\cdots\!21}{18\!\cdots\!53}a^{5}+\frac{54\!\cdots\!43}{18\!\cdots\!53}a^{4}+\frac{61\!\cdots\!45}{18\!\cdots\!53}a^{3}-\frac{27\!\cdots\!95}{18\!\cdots\!53}a^{2}+\frac{49\!\cdots\!48}{18\!\cdots\!53}a-\frac{43\!\cdots\!41}{19\!\cdots\!49}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{203}$, which has order $203$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $203$ (assuming GRH)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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 $6$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{500894425233379}{18\!\cdots\!53}a^{13}+\frac{9083208803277}{19\!\cdots\!49}a^{12}+\frac{10\!\cdots\!04}{18\!\cdots\!53}a^{11}-\frac{48\!\cdots\!58}{18\!\cdots\!53}a^{10}+\frac{13\!\cdots\!29}{18\!\cdots\!53}a^{9}-\frac{30\!\cdots\!04}{18\!\cdots\!53}a^{8}+\frac{69\!\cdots\!92}{18\!\cdots\!53}a^{7}-\frac{14\!\cdots\!59}{18\!\cdots\!53}a^{6}+\frac{24\!\cdots\!59}{18\!\cdots\!53}a^{5}+\frac{14\!\cdots\!04}{18\!\cdots\!53}a^{4}+\frac{31\!\cdots\!67}{18\!\cdots\!53}a^{3}+\frac{41\!\cdots\!43}{18\!\cdots\!53}a^{2}+\frac{24\!\cdots\!42}{18\!\cdots\!53}a-\frac{16\!\cdots\!09}{19\!\cdots\!49}$, $\frac{1182799648905}{19\!\cdots\!49}a^{13}+\frac{8568865761786}{19\!\cdots\!49}a^{12}+\frac{7684286930910}{19\!\cdots\!49}a^{11}+\frac{77847407861811}{19\!\cdots\!49}a^{10}-\frac{326021747139292}{19\!\cdots\!49}a^{9}+\frac{21\!\cdots\!20}{19\!\cdots\!49}a^{8}-\frac{59\!\cdots\!51}{19\!\cdots\!49}a^{7}+\frac{10\!\cdots\!52}{19\!\cdots\!49}a^{6}-\frac{27\!\cdots\!81}{19\!\cdots\!49}a^{5}+\frac{43\!\cdots\!62}{19\!\cdots\!49}a^{4}-\frac{86\!\cdots\!40}{19\!\cdots\!49}a^{3}+\frac{47\!\cdots\!64}{19\!\cdots\!49}a^{2}-\frac{27\!\cdots\!47}{19\!\cdots\!49}a-\frac{14\!\cdots\!79}{19\!\cdots\!49}$, $\frac{339974479466749}{18\!\cdots\!53}a^{13}+\frac{5231873784525}{19\!\cdots\!49}a^{12}+\frac{64\!\cdots\!91}{18\!\cdots\!53}a^{11}-\frac{46\!\cdots\!05}{18\!\cdots\!53}a^{10}+\frac{87\!\cdots\!95}{18\!\cdots\!53}a^{9}+\frac{19\!\cdots\!79}{18\!\cdots\!53}a^{8}+\frac{35\!\cdots\!75}{18\!\cdots\!53}a^{7}+\frac{17\!\cdots\!34}{18\!\cdots\!53}a^{6}+\frac{16\!\cdots\!17}{18\!\cdots\!53}a^{5}+\frac{10\!\cdots\!11}{18\!\cdots\!53}a^{4}+\frac{24\!\cdots\!84}{18\!\cdots\!53}a^{3}+\frac{47\!\cdots\!73}{18\!\cdots\!53}a^{2}+\frac{49\!\cdots\!29}{18\!\cdots\!53}a+\frac{45\!\cdots\!05}{19\!\cdots\!49}$, $\frac{8195604655819}{19\!\cdots\!49}a^{13}+\frac{7724519215672}{19\!\cdots\!49}a^{12}+\frac{143285362101380}{19\!\cdots\!49}a^{11}-\frac{276452985460239}{19\!\cdots\!49}a^{10}+\frac{18\!\cdots\!69}{19\!\cdots\!49}a^{9}-\frac{23\!\cdots\!44}{19\!\cdots\!49}a^{8}+\frac{70\!\cdots\!03}{19\!\cdots\!49}a^{7}-\frac{13\!\cdots\!61}{19\!\cdots\!49}a^{6}+\frac{22\!\cdots\!81}{19\!\cdots\!49}a^{5}-\frac{27\!\cdots\!18}{19\!\cdots\!49}a^{4}+\frac{31\!\cdots\!55}{19\!\cdots\!49}a^{3}-\frac{30\!\cdots\!10}{19\!\cdots\!49}a^{2}+\frac{18\!\cdots\!25}{19\!\cdots\!49}a-\frac{28\!\cdots\!54}{19\!\cdots\!49}$, $\frac{18709735634464}{19\!\cdots\!49}a^{13}+\frac{5140286677554}{19\!\cdots\!49}a^{12}+\frac{344302418627400}{19\!\cdots\!49}a^{11}-\frac{786930198666752}{19\!\cdots\!49}a^{10}+\frac{52\!\cdots\!31}{19\!\cdots\!49}a^{9}-\frac{84\!\cdots\!64}{19\!\cdots\!49}a^{8}+\frac{25\!\cdots\!96}{19\!\cdots\!49}a^{7}-\frac{42\!\cdots\!12}{19\!\cdots\!49}a^{6}+\frac{92\!\cdots\!84}{19\!\cdots\!49}a^{5}-\frac{12\!\cdots\!64}{19\!\cdots\!49}a^{4}+\frac{98\!\cdots\!23}{19\!\cdots\!49}a^{3}-\frac{13\!\cdots\!32}{19\!\cdots\!49}a^{2}+\frac{82\!\cdots\!56}{19\!\cdots\!49}a-\frac{10\!\cdots\!06}{19\!\cdots\!49}$, $\frac{957311663154}{19\!\cdots\!49}a^{13}-\frac{1382482426626}{19\!\cdots\!49}a^{12}+\frac{20501734780765}{19\!\cdots\!49}a^{11}-\frac{66405155977958}{19\!\cdots\!49}a^{10}+\frac{400067065916014}{19\!\cdots\!49}a^{9}-\frac{976206211656754}{19\!\cdots\!49}a^{8}+\frac{28\!\cdots\!12}{19\!\cdots\!49}a^{7}-\frac{48\!\cdots\!14}{19\!\cdots\!49}a^{6}+\frac{11\!\cdots\!70}{19\!\cdots\!49}a^{5}-\frac{16\!\cdots\!16}{19\!\cdots\!49}a^{4}+\frac{29\!\cdots\!62}{19\!\cdots\!49}a^{3}-\frac{18\!\cdots\!21}{19\!\cdots\!49}a^{2}+\frac{10\!\cdots\!53}{19\!\cdots\!49}a+\frac{23\!\cdots\!70}{19\!\cdots\!49}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 35256.6897369 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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)
# 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))))
# 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)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(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);
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)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(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);
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
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| 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.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
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.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.14.7.2 | $x^{14} + 21 x^{12} + 189 x^{10} + 4 x^{9} + 945 x^{8} - 94 x^{7} + 2835 x^{6} - 630 x^{5} + 5107 x^{4} + 630 x^{3} + 5131 x^{2} + 1242 x + 2212$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
|
\(7\)
| 7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |
| 7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.49.7t1.a.a | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.147.14t1.a.d | $1$ | $ 3 \cdot 7^{2}$ | 14.0.418988153029298748294987.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
| * | 1.49.7t1.a.b | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.147.14t1.a.f | $1$ | $ 3 \cdot 7^{2}$ | 14.0.418988153029298748294987.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
| * | 1.49.7t1.a.c | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.147.14t1.a.b | $1$ | $ 3 \cdot 7^{2}$ | 14.0.418988153029298748294987.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
| * | 1.49.7t1.a.d | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.147.14t1.a.c | $1$ | $ 3 \cdot 7^{2}$ | 14.0.418988153029298748294987.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
| * | 1.49.7t1.a.e | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.147.14t1.a.e | $1$ | $ 3 \cdot 7^{2}$ | 14.0.418988153029298748294987.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
| * | 1.49.7t1.a.f | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 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 *.