# Properties

 Label 16.0.731086699811838561.1 Degree $16$ Signature $[0, 8]$ Discriminant $7.311\times 10^{17}$ Root discriminant $$13.08$$ Ramified primes $3,19$ Class number $1$ Class group trivial Galois group $\SL(2,3):C_2$ (as 16T60)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 1)

gp: K = bnfinit(y^16 - 3*y^15 + 6*y^14 - 4*y^13 - y^12 + 15*y^11 - 27*y^10 + 32*y^9 - 12*y^8 - 23*y^7 + 55*y^6 - 57*y^5 + 43*y^4 - 23*y^3 + 9*y^2 - y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 1)

$$x^{16} - 3 x^{15} + 6 x^{14} - 4 x^{13} - x^{12} + 15 x^{11} - 27 x^{10} + 32 x^{9} - 12 x^{8} - 23 x^{7} + 55 x^{6} - 57 x^{5} + 43 x^{4} - 23 x^{3} + 9 x^{2} - x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

## Invariants

 Degree: $16$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K);  oscar: degree(K) Signature: $[0, 8]$ sage: K.signature()  gp: K.sign  magma: Signature(K);  oscar: signature(K) Discriminant: $$731086699811838561$$ 731086699811838561 $$\medspace = 3^{16}\cdot 19^{8}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$13.08$$ 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)) Ramified primes: $$3$$, $$19$$ 3, 19 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q$$ $\card{ \Aut(K/\Q) }$: $4$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{56195686}a^{15}-\frac{5048488}{28097843}a^{14}+\frac{1111075}{56195686}a^{13}+\frac{19107259}{56195686}a^{12}+\frac{15673121}{56195686}a^{11}-\frac{9795169}{56195686}a^{10}-\frac{12554447}{56195686}a^{9}-\frac{8559929}{28097843}a^{8}-\frac{983625}{56195686}a^{7}+\frac{1446632}{28097843}a^{6}-\frac{10805966}{28097843}a^{5}-\frac{4099715}{56195686}a^{4}+\frac{7014238}{28097843}a^{3}-\frac{5619997}{28097843}a^{2}-\frac{25527443}{56195686}a+\frac{7866912}{28097843}$

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 $1$

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: $7$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K);  oscar: rank(UK) Torsion generator: $$-1$$ -1  (order $2$) 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: $\frac{7934539}{28097843}a^{15}-\frac{36032205}{56195686}a^{14}+\frac{29188960}{28097843}a^{13}-\frac{1849667}{56195686}a^{12}-\frac{52396327}{56195686}a^{11}+\frac{197843763}{56195686}a^{10}-\frac{276757383}{56195686}a^{9}+\frac{192830397}{56195686}a^{8}+\frac{42632706}{28097843}a^{7}-\frac{438318261}{56195686}a^{6}+\frac{278020437}{28097843}a^{5}-\frac{166639012}{28097843}a^{4}+\frac{191243773}{56195686}a^{3}+\frac{12604128}{28097843}a^{2}-\frac{48883008}{28097843}a+\frac{54187467}{56195686}$, $\frac{1439675}{28097843}a^{15}-\frac{1042436}{28097843}a^{14}+\frac{4796478}{28097843}a^{13}+\frac{5238337}{28097843}a^{12}+\frac{2871781}{28097843}a^{11}+\frac{26003980}{28097843}a^{10}+\frac{7394827}{28097843}a^{9}+\frac{31041491}{28097843}a^{8}+\frac{30965325}{28097843}a^{7}-\frac{4886335}{28097843}a^{6}+\frac{44439636}{28097843}a^{5}+\frac{3805798}{28097843}a^{4}+\frac{25713173}{28097843}a^{3}+\frac{2791552}{28097843}a^{2}+\frac{26599057}{28097843}a+\frac{11171576}{28097843}$, $\frac{4121951}{28097843}a^{15}-\frac{12822501}{28097843}a^{14}+\frac{16885383}{28097843}a^{13}+\frac{4184490}{28097843}a^{12}-\frac{44445150}{28097843}a^{11}+\frac{73137660}{28097843}a^{10}-\frac{90684178}{28097843}a^{9}+\frac{11130368}{28097843}a^{8}+\frac{120888211}{28097843}a^{7}-\frac{265121129}{28097843}a^{6}+\frac{212163936}{28097843}a^{5}+\frac{5175839}{28097843}a^{4}-\frac{192553993}{28097843}a^{3}+\frac{165988522}{28097843}a^{2}-\frac{99885883}{28097843}a+\frac{43348017}{28097843}$, $\frac{8124973}{28097843}a^{15}-\frac{19902589}{28097843}a^{14}+\frac{38887720}{28097843}a^{13}-\frac{13881849}{28097843}a^{12}-\frac{10397403}{28097843}a^{11}+\frac{111260698}{28097843}a^{10}-\frac{156722427}{28097843}a^{9}+\frac{184442567}{28097843}a^{8}-\frac{28984792}{28097843}a^{7}-\frac{171779177}{28097843}a^{6}+\frac{319111060}{28097843}a^{5}-\frac{305588939}{28097843}a^{4}+\frac{234144010}{28097843}a^{3}-\frac{139320586}{28097843}a^{2}+\frac{62539061}{28097843}a-\frac{1599563}{28097843}$, $\frac{10661733}{56195686}a^{15}-\frac{26264597}{28097843}a^{14}+\frac{102971233}{56195686}a^{13}-\frac{117173647}{56195686}a^{12}-\frac{17575617}{56195686}a^{11}+\frac{192924721}{56195686}a^{10}-\frac{556524227}{56195686}a^{9}+\frac{308198797}{28097843}a^{8}-\frac{413961979}{56195686}a^{7}-\frac{172922577}{28097843}a^{6}+\frac{525756757}{28097843}a^{5}-\frac{1230722353}{56195686}a^{4}+\frac{455891920}{28097843}a^{3}-\frac{257277301}{28097843}a^{2}+\frac{164996481}{56195686}a-\frac{12056490}{28097843}$, $\frac{7970339}{56195686}a^{15}+\frac{1521107}{28097843}a^{14}-\frac{8969571}{56195686}a^{13}+\frac{67204139}{56195686}a^{12}-\frac{244309}{56195686}a^{11}+\frac{50805557}{56195686}a^{10}+\frac{132444845}{56195686}a^{9}-\frac{67541754}{28097843}a^{8}+\frac{260237729}{56195686}a^{7}+\frac{888297}{28097843}a^{6}-\frac{102301823}{28097843}a^{5}+\frac{380554837}{56195686}a^{4}-\frac{86356302}{28097843}a^{3}+\frac{47161030}{28097843}a^{2}-\frac{48998833}{56195686}a-\frac{10806226}{28097843}$, $\frac{7934539}{28097843}a^{15}-\frac{36032205}{56195686}a^{14}+\frac{29188960}{28097843}a^{13}-\frac{1849667}{56195686}a^{12}-\frac{52396327}{56195686}a^{11}+\frac{197843763}{56195686}a^{10}-\frac{276757383}{56195686}a^{9}+\frac{192830397}{56195686}a^{8}+\frac{42632706}{28097843}a^{7}-\frac{438318261}{56195686}a^{6}+\frac{278020437}{28097843}a^{5}-\frac{166639012}{28097843}a^{4}+\frac{191243773}{56195686}a^{3}+\frac{12604128}{28097843}a^{2}-\frac{48883008}{28097843}a-\frac{2008219}{56195686}$ 7934539/28097843*a^15 - 36032205/56195686*a^14 + 29188960/28097843*a^13 - 1849667/56195686*a^12 - 52396327/56195686*a^11 + 197843763/56195686*a^10 - 276757383/56195686*a^9 + 192830397/56195686*a^8 + 42632706/28097843*a^7 - 438318261/56195686*a^6 + 278020437/28097843*a^5 - 166639012/28097843*a^4 + 191243773/56195686*a^3 + 12604128/28097843*a^2 - 48883008/28097843*a + 54187467/56195686, 1439675/28097843*a^15 - 1042436/28097843*a^14 + 4796478/28097843*a^13 + 5238337/28097843*a^12 + 2871781/28097843*a^11 + 26003980/28097843*a^10 + 7394827/28097843*a^9 + 31041491/28097843*a^8 + 30965325/28097843*a^7 - 4886335/28097843*a^6 + 44439636/28097843*a^5 + 3805798/28097843*a^4 + 25713173/28097843*a^3 + 2791552/28097843*a^2 + 26599057/28097843*a + 11171576/28097843, 4121951/28097843*a^15 - 12822501/28097843*a^14 + 16885383/28097843*a^13 + 4184490/28097843*a^12 - 44445150/28097843*a^11 + 73137660/28097843*a^10 - 90684178/28097843*a^9 + 11130368/28097843*a^8 + 120888211/28097843*a^7 - 265121129/28097843*a^6 + 212163936/28097843*a^5 + 5175839/28097843*a^4 - 192553993/28097843*a^3 + 165988522/28097843*a^2 - 99885883/28097843*a + 43348017/28097843, 8124973/28097843*a^15 - 19902589/28097843*a^14 + 38887720/28097843*a^13 - 13881849/28097843*a^12 - 10397403/28097843*a^11 + 111260698/28097843*a^10 - 156722427/28097843*a^9 + 184442567/28097843*a^8 - 28984792/28097843*a^7 - 171779177/28097843*a^6 + 319111060/28097843*a^5 - 305588939/28097843*a^4 + 234144010/28097843*a^3 - 139320586/28097843*a^2 + 62539061/28097843*a - 1599563/28097843, 10661733/56195686*a^15 - 26264597/28097843*a^14 + 102971233/56195686*a^13 - 117173647/56195686*a^12 - 17575617/56195686*a^11 + 192924721/56195686*a^10 - 556524227/56195686*a^9 + 308198797/28097843*a^8 - 413961979/56195686*a^7 - 172922577/28097843*a^6 + 525756757/28097843*a^5 - 1230722353/56195686*a^4 + 455891920/28097843*a^3 - 257277301/28097843*a^2 + 164996481/56195686*a - 12056490/28097843, 7970339/56195686*a^15 + 1521107/28097843*a^14 - 8969571/56195686*a^13 + 67204139/56195686*a^12 - 244309/56195686*a^11 + 50805557/56195686*a^10 + 132444845/56195686*a^9 - 67541754/28097843*a^8 + 260237729/56195686*a^7 + 888297/28097843*a^6 - 102301823/28097843*a^5 + 380554837/56195686*a^4 - 86356302/28097843*a^3 + 47161030/28097843*a^2 - 48998833/56195686*a - 10806226/28097843, 7934539/28097843*a^15 - 36032205/56195686*a^14 + 29188960/28097843*a^13 - 1849667/56195686*a^12 - 52396327/56195686*a^11 + 197843763/56195686*a^10 - 276757383/56195686*a^9 + 192830397/56195686*a^8 + 42632706/28097843*a^7 - 438318261/56195686*a^6 + 278020437/28097843*a^5 - 166639012/28097843*a^4 + 191243773/56195686*a^3 + 12604128/28097843*a^2 - 48883008/28097843*a - 2008219/56195686 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: $$188.5109795655095$$ 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)^{8}\cdot 188.5109795655095 \cdot 1}{2\cdot\sqrt{731086699811838561}}\cr\approx \mathstrut & 0.267769532159754 \end{aligned}

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 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^16 - 3*x^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 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^16 - 3*x^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 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^16 - 3*x^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 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

$\SL(2,3):C_2$ (as 16T60):

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 48 The 14 conjugacy class representatives for $\SL(2,3):C_2$ Character table for $\SL(2,3):C_2$

## Intermediate fields

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 24 sibling: deg 24 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 ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ R ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ R ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$

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$LabelPolynomial$efc$Galois group Slope content $$3$$ 3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{4} 3.12.16.14x^{12} + 24 x^{11} + 216 x^{10} + 768 x^{9} - 432 x^{8} - 10368 x^{7} - 18414 x^{6} + 27864 x^{5} + 83592 x^{4} + 10800 x^{3} + 64800 x^{2} + 901125$$3$$4$$16$$C_{12}$$[2]^{4}$$$19$$ 19.8.4.1$x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4} 19.8.4.1x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$## Artin representations Label Dimension Conductor Artin stem field$G$Ind$\chi(c)$* 1.1.1t1.a.a$11$$$\Q$$$C_111$* 1.19.2t1.a.a$1 19 $$$\Q(\sqrt{-19})$$$C_2$(as 2T1)$1-1$1.9.3t1.a.a$1 3^{2}$$$\Q(\zeta_{9})^+$$$C_3$(as 3T1)$01$1.171.6t1.f.a$1 3^{2} \cdot 19 $6.0.45001899.1$C_6$(as 6T1)$0-1$1.9.3t1.a.b$1 3^{2}$$$\Q(\zeta_{9})^+$$$C_3$(as 3T1)$01$1.171.6t1.f.b$1 3^{2} \cdot 19 $6.0.45001899.1$C_6$(as 6T1)$0-1$2.1539.24t21.a.a$2 3^{4} \cdot 19 $16.0.731086699811838561.1$\SL(2,3):C_2$(as 16T60)$00$2.1539.24t21.a.b$2 3^{4} \cdot 19 $16.0.731086699811838561.1$\SL(2,3):C_2$(as 16T60)$00$* 2.171.16t60.a.a$2 3^{2} \cdot 19 $16.0.731086699811838561.1$\SL(2,3):C_2$(as 16T60)$00$* 2.171.16t60.a.b$2 3^{2} \cdot 19 $16.0.731086699811838561.1$\SL(2,3):C_2$(as 16T60)$00$* 2.171.16t60.a.c$2 3^{2} \cdot 19 $16.0.731086699811838561.1$\SL(2,3):C_2$(as 16T60)$00$* 2.171.16t60.a.d$2 3^{2} \cdot 19 $16.0.731086699811838561.1$\SL(2,3):C_2$(as 16T60)$00$* 3.1539.6t6.a.a$3 3^{4} \cdot 19 $6.4.124659.1$A_4\times C_2$(as 6T6)$11$* 3.29241.4t4.b.a$3 3^{4} \cdot 19^{2}$4.0.29241.1$A_4$(as 4T4)$1-1\$

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 *.