Properties

Label 16.0.308...281.2
Degree $16$
Signature $[0, 8]$
Discriminant $3.085\times 10^{19}$
Root discriminant \(16.52\)
Ramified primes $3,7,13$
Class number $1$
Class group trivial
Galois group $\SL(2,3):C_2$ (as 16T60)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 31*x^13 - 8*x^12 - 101*x^11 + 7*x^10 + 183*x^9 + 32*x^8 - 183*x^7 - 61*x^6 + 112*x^5 + 78*x^4 - 9*x^3 - 28*x^2 - 7*x + 7)
 
gp: K = bnfinit(y^16 - 5*y^15 + 31*y^13 - 8*y^12 - 101*y^11 + 7*y^10 + 183*y^9 + 32*y^8 - 183*y^7 - 61*y^6 + 112*y^5 + 78*y^4 - 9*y^3 - 28*y^2 - 7*y + 7, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 31*x^13 - 8*x^12 - 101*x^11 + 7*x^10 + 183*x^9 + 32*x^8 - 183*x^7 - 61*x^6 + 112*x^5 + 78*x^4 - 9*x^3 - 28*x^2 - 7*x + 7);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 + 31*x^13 - 8*x^12 - 101*x^11 + 7*x^10 + 183*x^9 + 32*x^8 - 183*x^7 - 61*x^6 + 112*x^5 + 78*x^4 - 9*x^3 - 28*x^2 - 7*x + 7)
 

\( x^{16} - 5 x^{15} + 31 x^{13} - 8 x^{12} - 101 x^{11} + 7 x^{10} + 183 x^{9} + 32 x^{8} - 183 x^{7} - 61 x^{6} + 112 x^{5} + 78 x^{4} - 9 x^{3} - 28 x^{2} - 7 x + 7 \) Copy content Toggle raw display

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:   \(30853268336830129281\) \(\medspace = 3^{8}\cdot 7^{8}\cdot 13^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(16.52\)
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:  $3^{1/2}7^{2/3}13^{2/3}\approx 35.04194650073235$
Ramified primes:   \(3\), \(7\), \(13\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7}a^{13}+\frac{1}{7}a^{12}+\frac{2}{7}a^{11}+\frac{3}{7}a^{10}+\frac{1}{7}a^{9}+\frac{3}{7}a^{8}-\frac{3}{7}a^{7}-\frac{2}{7}a^{6}+\frac{1}{7}a^{5}+\frac{2}{7}a^{4}-\frac{2}{7}a^{3}$, $\frac{1}{14}a^{14}+\frac{1}{14}a^{12}-\frac{3}{7}a^{11}+\frac{5}{14}a^{10}+\frac{1}{7}a^{9}-\frac{3}{7}a^{8}+\frac{1}{14}a^{7}+\frac{3}{14}a^{6}+\frac{1}{14}a^{5}+\frac{3}{14}a^{4}+\frac{1}{7}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{201188162}a^{15}-\frac{2928463}{100594081}a^{14}+\frac{8434785}{201188162}a^{13}+\frac{28114571}{100594081}a^{12}-\frac{98515681}{201188162}a^{11}+\frac{25976444}{100594081}a^{10}-\frac{7398143}{100594081}a^{9}-\frac{59162769}{201188162}a^{8}-\frac{810261}{1991962}a^{7}-\frac{38192121}{201188162}a^{6}-\frac{20136635}{201188162}a^{5}-\frac{9431326}{100594081}a^{4}+\frac{87385559}{201188162}a^{3}+\frac{2754567}{14370583}a^{2}+\frac{9175725}{28741166}a+\frac{177689}{14370583}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $7$

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:   \( \frac{2412144}{14370583} a^{15} - \frac{14391461}{14370583} a^{14} + \frac{15876474}{14370583} a^{13} + \frac{49247605}{14370583} a^{12} - \frac{65201610}{14370583} a^{11} - \frac{117821061}{14370583} a^{10} + \frac{93665114}{14370583} a^{9} + \frac{169284546}{14370583} a^{8} - \frac{48574}{142283} a^{7} - \frac{146349640}{14370583} a^{6} - \frac{65202436}{14370583} a^{5} + \frac{89113643}{14370583} a^{4} + \frac{104995800}{14370583} a^{3} + \frac{10704428}{14370583} a^{2} + \frac{3108868}{14370583} a - \frac{5893873}{14370583} \)  (order $6$) Copy content Toggle raw display
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:   $\frac{46989633}{100594081}a^{15}-\frac{39451927}{14370583}a^{14}+\frac{233828640}{100594081}a^{13}+\frac{1311067369}{100594081}a^{12}-\frac{234728392}{14370583}a^{11}-\frac{3385791622}{100594081}a^{10}+\frac{3730221253}{100594081}a^{9}+\frac{5247556999}{100594081}a^{8}-\frac{5336050}{142283}a^{7}-\frac{4806752718}{100594081}a^{6}+\frac{1681630089}{100594081}a^{5}+\frac{3114844872}{100594081}a^{4}+\frac{944429679}{100594081}a^{3}-\frac{115289247}{14370583}a^{2}-\frac{86059412}{14370583}a+\frac{18862201}{14370583}$, $\frac{24630651}{100594081}a^{15}-\frac{269122433}{201188162}a^{14}+\frac{49257671}{100594081}a^{13}+\frac{1664320717}{201188162}a^{12}-\frac{744423399}{100594081}a^{11}-\frac{4720848071}{201188162}a^{10}+\frac{2007668504}{100594081}a^{9}+\frac{3819051272}{100594081}a^{8}-\frac{49785665}{1991962}a^{7}-\frac{6781573229}{201188162}a^{6}+\frac{3286029769}{201188162}a^{5}+\frac{576310713}{28741166}a^{4}+\frac{69506036}{100594081}a^{3}-\frac{124722977}{28741166}a^{2}-\frac{38071239}{14370583}a+\frac{66918641}{28741166}$, $\frac{8251872}{100594081}a^{15}-\frac{81856967}{201188162}a^{14}-\frac{2946340}{100594081}a^{13}+\frac{549009973}{201188162}a^{12}-\frac{140187044}{100594081}a^{11}-\frac{1443641963}{201188162}a^{10}+\frac{171683581}{100594081}a^{9}+\frac{1148978596}{100594081}a^{8}+\frac{1148229}{1991962}a^{7}-\frac{1670939181}{201188162}a^{6}-\frac{580748745}{201188162}a^{5}+\frac{860579271}{201188162}a^{4}+\frac{364009693}{100594081}a^{3}+\frac{52503247}{28741166}a^{2}-\frac{12261589}{14370583}a-\frac{12875141}{28741166}$, $\frac{25369882}{100594081}a^{15}-\frac{286934267}{201188162}a^{14}+\frac{11470186}{14370583}a^{13}+\frac{1641051961}{201188162}a^{12}-\frac{858188525}{100594081}a^{11}-\frac{4541461495}{201188162}a^{10}+\frac{2217399390}{100594081}a^{9}+\frac{527842621}{14370583}a^{8}-\frac{52963179}{1991962}a^{7}-\frac{6854141879}{201188162}a^{6}+\frac{3343246171}{201188162}a^{5}+\frac{4275719891}{201188162}a^{4}+\frac{70681136}{100594081}a^{3}-\frac{210930805}{28741166}a^{2}-\frac{59103262}{14370583}a+\frac{62531631}{28741166}$, $\frac{15306591}{100594081}a^{15}-\frac{11814331}{14370583}a^{14}+\frac{14755659}{100594081}a^{13}+\frac{592165758}{100594081}a^{12}-\frac{550357996}{100594081}a^{11}-\frac{241567021}{14370583}a^{10}+\frac{1716211177}{100594081}a^{9}+\frac{2716985790}{100594081}a^{8}-\frac{24887279}{995981}a^{7}-\frac{342534968}{14370583}a^{6}+\frac{283710550}{14370583}a^{5}+\frac{1326600139}{100594081}a^{4}-\frac{528375770}{100594081}a^{3}-\frac{31541984}{14370583}a^{2}-\frac{12553567}{14370583}a+\frac{15136906}{14370583}$, $\frac{34969023}{201188162}a^{15}-\frac{267070075}{201188162}a^{14}+\frac{606895707}{201188162}a^{13}+\frac{162331049}{201188162}a^{12}-\frac{1977265495}{201188162}a^{11}+\frac{575609287}{201188162}a^{10}+\frac{235879479}{14370583}a^{9}-\frac{1610063225}{201188162}a^{8}-\frac{12079496}{995981}a^{7}+\frac{587031092}{100594081}a^{6}+\frac{409350481}{100594081}a^{5}-\frac{289581377}{201188162}a^{4}+\frac{210357975}{201188162}a^{3}-\frac{27431237}{28741166}a^{2}+\frac{48754335}{28741166}a-\frac{41975799}{28741166}$, $\frac{25843939}{201188162}a^{15}-\frac{127079777}{201188162}a^{14}+\frac{814733}{28741166}a^{13}+\frac{723526581}{201188162}a^{12}-\frac{23463341}{28741166}a^{11}-\frac{2096540653}{201188162}a^{10}-\frac{56012529}{100594081}a^{9}+\frac{3101679557}{201188162}a^{8}+\frac{5856348}{995981}a^{7}-\frac{985955228}{100594081}a^{6}-\frac{637940836}{100594081}a^{5}+\frac{716074859}{201188162}a^{4}+\frac{1192809365}{201188162}a^{3}+\frac{94172601}{28741166}a^{2}+\frac{69500623}{28741166}a+\frac{55905727}{28741166}$ Copy content Toggle raw display
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:  \( 7145.924896370814 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 7145.924896370814 \cdot 1}{6\cdot\sqrt{30853268336830129281}}\cr\approx \mathstrut & 0.520829477092895 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 31*x^13 - 8*x^12 - 101*x^11 + 7*x^10 + 183*x^9 + 32*x^8 - 183*x^7 - 61*x^6 + 112*x^5 + 78*x^4 - 9*x^3 - 28*x^2 - 7*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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - 5*x^15 + 31*x^13 - 8*x^12 - 101*x^11 + 7*x^10 + 183*x^9 + 32*x^8 - 183*x^7 - 61*x^6 + 112*x^5 + 78*x^4 - 9*x^3 - 28*x^2 - 7*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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 5*x^15 + 31*x^13 - 8*x^12 - 101*x^11 + 7*x^10 + 183*x^9 + 32*x^8 - 183*x^7 - 61*x^6 + 112*x^5 + 78*x^4 - 9*x^3 - 28*x^2 - 7*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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 + 31*x^13 - 8*x^12 - 101*x^11 + 7*x^10 + 183*x^9 + 32*x^8 - 183*x^7 - 61*x^6 + 112*x^5 + 78*x^4 - 9*x^3 - 28*x^2 - 7*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

$\SL(2,3):C_2$ (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

\(\Q(\sqrt{-3}) \), 4.0.74529.1, 8.0.5554571841.6

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 ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ R ${\href{/padicField/11.2.0.1}{2} }^{8}$ R ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$

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 $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
\(13\) Copy content Toggle raw display 13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.6.4.3$x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$$3$$2$$4$$C_6$$[\ ]_{3}^{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) $1$ $-1$
1.91.3t1.a.a$1$ $ 7 \cdot 13 $ 3.3.8281.2 $C_3$ (as 3T1) $0$ $1$
1.273.6t1.d.a$1$ $ 3 \cdot 7 \cdot 13 $ 6.0.1851523947.1 $C_6$ (as 6T1) $0$ $-1$
1.91.3t1.a.b$1$ $ 7 \cdot 13 $ 3.3.8281.2 $C_3$ (as 3T1) $0$ $1$
1.273.6t1.d.b$1$ $ 3 \cdot 7 \cdot 13 $ 6.0.1851523947.1 $C_6$ (as 6T1) $0$ $-1$
2.24843.24t21.a.a$2$ $ 3 \cdot 7^{2} \cdot 13^{2}$ 16.0.30853268336830129281.2 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
2.24843.24t21.a.b$2$ $ 3 \cdot 7^{2} \cdot 13^{2}$ 16.0.30853268336830129281.2 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.273.16t60.a.a$2$ $ 3 \cdot 7 \cdot 13 $ 16.0.30853268336830129281.2 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.273.16t60.a.b$2$ $ 3 \cdot 7 \cdot 13 $ 16.0.30853268336830129281.2 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.273.16t60.a.c$2$ $ 3 \cdot 7 \cdot 13 $ 16.0.30853268336830129281.2 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.273.16t60.a.d$2$ $ 3 \cdot 7 \cdot 13 $ 16.0.30853268336830129281.2 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 3.24843.6t6.a.a$3$ $ 3 \cdot 7^{2} \cdot 13^{2}$ 6.4.205724883.1 $A_4\times C_2$ (as 6T6) $1$ $1$
* 3.74529.4t4.a.a$3$ $ 3^{2} \cdot 7^{2} \cdot 13^{2}$ 4.0.74529.1 $A_4$ (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 *.