Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^21 - 2*x^15 + 9*x^12 - 8*x^9 + 5*x^6 - 3*x^3 + 1)
gp: K = bnfinit(y^24 - 2*y^21 - 2*y^15 + 9*y^12 - 8*y^9 + 5*y^6 - 3*y^3 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^21 - 2*x^15 + 9*x^12 - 8*x^9 + 5*x^6 - 3*x^3 + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - 2*x^21 - 2*x^15 + 9*x^12 - 8*x^9 + 5*x^6 - 3*x^3 + 1)
\( x^{24} - 2x^{21} - 2x^{15} + 9x^{12} - 8x^{9} + 5x^{6} - 3x^{3} + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(572565594852444156646728515625\)
\(\medspace = 3^{36}\cdot 5^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.37\) | 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: | $3^{31/18}5^{3/4}\approx 22.178712478478406$ | ||
| Ramified primes: |
\(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{61}a^{21}-\frac{26}{61}a^{18}+\frac{14}{61}a^{15}+\frac{28}{61}a^{12}+\frac{8}{61}a^{9}-\frac{17}{61}a^{6}-\frac{14}{61}a^{3}+\frac{28}{61}$, $\frac{1}{61}a^{22}-\frac{26}{61}a^{19}+\frac{14}{61}a^{16}+\frac{28}{61}a^{13}+\frac{8}{61}a^{10}-\frac{17}{61}a^{7}-\frac{14}{61}a^{4}+\frac{28}{61}a$, $\frac{1}{61}a^{23}-\frac{26}{61}a^{20}+\frac{14}{61}a^{17}+\frac{28}{61}a^{14}+\frac{8}{61}a^{11}-\frac{17}{61}a^{8}-\frac{14}{61}a^{5}+\frac{28}{61}a^{2}$
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
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{19}{61} a^{21} + \frac{6}{61} a^{18} + \frac{39}{61} a^{15} + \frac{78}{61} a^{12} - \frac{91}{61} a^{9} - \frac{104}{61} a^{6} - \frac{39}{61} a^{3} + \frac{17}{61} \)
(order $30$)
| 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: |
$\frac{92}{61}a^{23}-\frac{135}{61}a^{20}-\frac{54}{61}a^{17}-\frac{230}{61}a^{14}+\frac{675}{61}a^{11}-\frac{405}{61}a^{8}+\frac{359}{61}a^{5}-\frac{108}{61}a^{2}$, $\frac{6}{61}a^{22}-\frac{34}{61}a^{19}+\frac{23}{61}a^{16}+\frac{46}{61}a^{13}+\frac{109}{61}a^{10}-\frac{224}{61}a^{7}-\frac{23}{61}a^{4}+\frac{46}{61}a$, $\frac{17}{61}a^{23}-\frac{15}{61}a^{20}-\frac{6}{61}a^{17}-\frac{73}{61}a^{14}+\frac{75}{61}a^{11}-\frac{45}{61}a^{8}+\frac{189}{61}a^{5}-\frac{73}{61}a^{2}$, $\frac{96}{61}a^{23}-\frac{50}{61}a^{21}-\frac{117}{61}a^{20}+\frac{80}{61}a^{18}-\frac{120}{61}a^{17}+\frac{32}{61}a^{15}-\frac{240}{61}a^{14}+\frac{125}{61}a^{12}+\frac{707}{61}a^{11}-\frac{400}{61}a^{9}-\frac{168}{61}a^{8}+\frac{240}{61}a^{6}+\frac{120}{61}a^{5}-\frac{215}{61}a^{3}-\frac{118}{61}a^{2}+\frac{125}{61}$, $\frac{17}{61}a^{22}+\frac{50}{61}a^{21}-\frac{15}{61}a^{19}-\frac{80}{61}a^{18}-\frac{6}{61}a^{16}-\frac{32}{61}a^{15}-\frac{73}{61}a^{13}-\frac{125}{61}a^{12}+\frac{75}{61}a^{10}+\frac{400}{61}a^{9}-\frac{45}{61}a^{7}-\frac{240}{61}a^{6}+\frac{128}{61}a^{4}+\frac{215}{61}a^{3}-\frac{12}{61}a-\frac{64}{61}$, $\frac{17}{61}a^{23}+\frac{75}{61}a^{21}-\frac{15}{61}a^{20}-\frac{120}{61}a^{18}-\frac{6}{61}a^{17}-\frac{48}{61}a^{15}-\frac{73}{61}a^{14}-\frac{157}{61}a^{12}+\frac{75}{61}a^{11}+\frac{600}{61}a^{9}-\frac{45}{61}a^{8}-\frac{360}{61}a^{6}+\frac{189}{61}a^{5}+\frac{170}{61}a^{3}-\frac{12}{61}a^{2}-\frac{96}{61}$, $\frac{12}{61}a^{23}+\frac{64}{61}a^{21}-\frac{7}{61}a^{20}-\frac{78}{61}a^{18}-\frac{15}{61}a^{17}-\frac{80}{61}a^{15}-\frac{30}{61}a^{14}-\frac{160}{61}a^{12}+\frac{35}{61}a^{11}+\frac{451}{61}a^{9}-\frac{21}{61}a^{8}-\frac{112}{61}a^{6}+\frac{15}{61}a^{5}+\frac{80}{61}a^{3}+\frac{92}{61}a^{2}-\frac{38}{61}$, $\frac{12}{61}a^{23}+\frac{31}{61}a^{21}-\frac{68}{61}a^{20}-\frac{74}{61}a^{18}+\frac{46}{61}a^{17}+\frac{7}{61}a^{15}+\frac{31}{61}a^{14}-\frac{47}{61}a^{12}+\frac{218}{61}a^{11}+\frac{309}{61}a^{9}-\frac{387}{61}a^{8}-\frac{283}{61}a^{6}+\frac{137}{61}a^{5}+\frac{176}{61}a^{3}-\frac{91}{61}a^{2}-\frac{108}{61}$, $\frac{50}{61}a^{22}-\frac{80}{61}a^{19}-\frac{32}{61}a^{16}-\frac{125}{61}a^{13}+\frac{400}{61}a^{10}-\frac{240}{61}a^{7}+\frac{215}{61}a^{4}-\frac{64}{61}a+1$, $a^{23}+\frac{44}{61}a^{21}-2a^{20}-\frac{46}{61}a^{18}-\frac{55}{61}a^{15}-2a^{14}-\frac{110}{61}a^{12}+9a^{11}+\frac{291}{61}a^{9}-8a^{8}-\frac{77}{61}a^{6}+5a^{5}+\frac{55}{61}a^{3}-3a^{2}+\frac{12}{61}$, $\frac{96}{61}a^{23}+\frac{64}{61}a^{21}-\frac{117}{61}a^{20}-\frac{78}{61}a^{18}-\frac{120}{61}a^{17}-\frac{80}{61}a^{15}-\frac{240}{61}a^{14}-\frac{160}{61}a^{12}+\frac{707}{61}a^{11}+\frac{451}{61}a^{9}-\frac{168}{61}a^{8}-\frac{112}{61}a^{6}+\frac{120}{61}a^{5}+\frac{80}{61}a^{3}-\frac{57}{61}a^{2}-\frac{38}{61}$
| 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: | \( 1414385.0473488418 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 1414385.0473488418 \cdot 1}{30\cdot\sqrt{572565594852444156646728515625}}\cr\approx \mathstrut & 0.235880585985437 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^21 - 2*x^15 + 9*x^12 - 8*x^9 + 5*x^6 - 3*x^3 + 1)
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^24 - 2*x^21 - 2*x^15 + 9*x^12 - 8*x^9 + 5*x^6 - 3*x^3 + 1, 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^24 - 2*x^21 - 2*x^15 + 9*x^12 - 8*x^9 + 5*x^6 - 3*x^3 + 1);
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^24 - 2*x^21 - 2*x^15 + 9*x^12 - 8*x^9 + 5*x^6 - 3*x^3 + 1);
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
$S_3\times C_{12}$ (as 24T65):
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 solvable group of order 72 |
| The 36 conjugacy class representatives for $S_3\times C_{12}$ |
| Character table for $S_3\times C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.2460375.2, \(\Q(\zeta_{15})\), 12.0.6053445140625.2 |
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]
Sibling fields
| Degree 36 siblings: | data not computed |
| 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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.4.0.1}{4} }^{3}$ | R | R | ${\href{/padicField/7.12.0.1}{12} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.12.0.1}{12} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{12}$ | ${\href{/padicField/37.4.0.1}{4} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $24$ | $6$ | $4$ | $36$ | |||
|
\(5\)
| Deg $24$ | $4$ | $6$ | $18$ |
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) | $1$ | $-1$ |
| * | 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.45.6t1.b.a | $1$ | $ 3^{2} \cdot 5 $ | 6.0.2460375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.45.6t1.a.a | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.9.6t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.9.6t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.45.6t1.b.b | $1$ | $ 3^{2} \cdot 5 $ | 6.0.2460375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.45.6t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| * | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.15.4t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.15.4t1.a.b | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| 1.45.12t1.b.a | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.45.12t1.b.b | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.45.12t1.a.a | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.45.12t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.45.12t1.b.c | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.45.12t1.b.d | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.45.12t1.a.c | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.45.12t1.a.d | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 2.135.3t2.b.a | $2$ | $ 3^{3} \cdot 5 $ | 3.1.135.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.135.6t3.a.a | $2$ | $ 3^{3} \cdot 5 $ | 6.2.91125.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.405.6t5.a.a | $2$ | $ 3^{4} \cdot 5 $ | 6.0.2460375.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.405.12t18.a.a | $2$ | $ 3^{4} \cdot 5 $ | 12.0.6053445140625.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.405.6t5.a.b | $2$ | $ 3^{4} \cdot 5 $ | 6.0.2460375.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.405.12t18.a.b | $2$ | $ 3^{4} \cdot 5 $ | 12.0.6053445140625.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| 2.675.12t11.a.a | $2$ | $ 3^{3} \cdot 5^{2}$ | 12.0.1037970703125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| 2.675.12t11.a.b | $2$ | $ 3^{3} \cdot 5^{2}$ | 12.0.1037970703125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| * | 2.2025.24t65.a.a | $2$ | $ 3^{4} \cdot 5^{2}$ | 24.0.572565594852444156646728515625.2 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.2025.24t65.a.b | $2$ | $ 3^{4} \cdot 5^{2}$ | 24.0.572565594852444156646728515625.2 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.2025.24t65.a.c | $2$ | $ 3^{4} \cdot 5^{2}$ | 24.0.572565594852444156646728515625.2 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.2025.24t65.a.d | $2$ | $ 3^{4} \cdot 5^{2}$ | 24.0.572565594852444156646728515625.2 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
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 *.