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Properties

Label 30.2.119...125.1
Degree $30$
Signature $[2, 14]$
Discriminant $1.195\times 10^{49}$
Root discriminant $43.24$
Ramified primes $5, 571$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_{30}$ (as 30T14)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 5*x^29 + 11*x^28 - 4*x^27 - 19*x^26 + 37*x^25 + 129*x^24 - 1314*x^23 + 6565*x^22 - 21133*x^21 + 56331*x^20 - 116407*x^19 + 192164*x^18 - 249788*x^17 + 240591*x^16 - 123362*x^15 + 65684*x^14 - 91823*x^13 + 137991*x^12 - 356197*x^11 - 307772*x^10 + 22452*x^9 + 102955*x^8 + 347102*x^7 + 442668*x^6 - 57105*x^5 - 325253*x^4 - 81893*x^3 + 80109*x^2 + 46065*x + 6845)
 
gp: K = bnfinit(x^30 - 5*x^29 + 11*x^28 - 4*x^27 - 19*x^26 + 37*x^25 + 129*x^24 - 1314*x^23 + 6565*x^22 - 21133*x^21 + 56331*x^20 - 116407*x^19 + 192164*x^18 - 249788*x^17 + 240591*x^16 - 123362*x^15 + 65684*x^14 - 91823*x^13 + 137991*x^12 - 356197*x^11 - 307772*x^10 + 22452*x^9 + 102955*x^8 + 347102*x^7 + 442668*x^6 - 57105*x^5 - 325253*x^4 - 81893*x^3 + 80109*x^2 + 46065*x + 6845, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6845, 46065, 80109, -81893, -325253, -57105, 442668, 347102, 102955, 22452, -307772, -356197, 137991, -91823, 65684, -123362, 240591, -249788, 192164, -116407, 56331, -21133, 6565, -1314, 129, 37, -19, -4, 11, -5, 1]);
 

\( x^{30} - 5 x^{29} + 11 x^{28} - 4 x^{27} - 19 x^{26} + 37 x^{25} + 129 x^{24} - 1314 x^{23} + 6565 x^{22} - 21133 x^{21} + 56331 x^{20} - 116407 x^{19} + 192164 x^{18} - 249788 x^{17} + 240591 x^{16} - 123362 x^{15} + 65684 x^{14} - 91823 x^{13} + 137991 x^{12} - 356197 x^{11} - 307772 x^{10} + 22452 x^{9} + 102955 x^{8} + 347102 x^{7} + 442668 x^{6} - 57105 x^{5} - 325253 x^{4} - 81893 x^{3} + 80109 x^{2} + 46065 x + 6845 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(11952412704796787125889897969349006221954345703125\)\(\medspace = 5^{15}\cdot 571^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $43.24$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 571$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} + \frac{1}{5} a^{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{6} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{7} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{8} + \frac{2}{5} a^{4}$, $\frac{1}{25} a^{17} + \frac{2}{25} a^{16} - \frac{1}{25} a^{15} - \frac{2}{25} a^{14} + \frac{2}{25} a^{13} - \frac{1}{25} a^{12} - \frac{2}{25} a^{11} + \frac{1}{25} a^{10} + \frac{8}{25} a^{9} - \frac{4}{25} a^{8} + \frac{12}{25} a^{7} - \frac{11}{25} a^{6} - \frac{6}{25} a^{5} - \frac{7}{25} a^{4} - \frac{9}{25} a^{3} + \frac{12}{25} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{25} a^{18} + \frac{1}{25} a^{14} + \frac{1}{25} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{9}{25} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{6}{25} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{25} a^{19} + \frac{1}{25} a^{15} + \frac{1}{25} a^{11} + \frac{1}{5} a^{9} - \frac{9}{25} a^{7} - \frac{2}{5} a^{5} + \frac{6}{25} a^{3} + \frac{1}{5} a$, $\frac{1}{25} a^{20} + \frac{1}{25} a^{16} + \frac{1}{25} a^{12} - \frac{9}{25} a^{8} + \frac{6}{25} a^{4}$, $\frac{1}{125} a^{21} - \frac{2}{125} a^{20} - \frac{1}{125} a^{19} + \frac{2}{125} a^{18} + \frac{1}{125} a^{17} - \frac{12}{125} a^{16} + \frac{9}{125} a^{15} - \frac{3}{125} a^{14} + \frac{6}{125} a^{13} - \frac{12}{125} a^{12} + \frac{4}{125} a^{11} - \frac{3}{125} a^{10} - \frac{9}{125} a^{9} + \frac{28}{125} a^{8} + \frac{19}{125} a^{7} - \frac{43}{125} a^{6} + \frac{41}{125} a^{5} + \frac{13}{125} a^{4} - \frac{6}{125} a^{3} - \frac{3}{125} a^{2} + \frac{12}{25} a + \frac{7}{25}$, $\frac{1}{125} a^{22} + \frac{2}{25} a^{16} + \frac{1}{25} a^{15} - \frac{1}{25} a^{13} + \frac{2}{25} a^{11} - \frac{2}{25} a^{10} + \frac{8}{25} a^{9} + \frac{8}{25} a^{8} - \frac{12}{25} a^{7} - \frac{12}{25} a^{6} - \frac{8}{25} a^{5} + \frac{11}{25} a^{4} + \frac{9}{25} a^{3} - \frac{56}{125} a^{2} + \frac{1}{25} a + \frac{4}{25}$, $\frac{1}{125} a^{23} + \frac{2}{25} a^{16} + \frac{2}{25} a^{15} - \frac{2}{25} a^{14} + \frac{1}{25} a^{13} - \frac{1}{25} a^{12} + \frac{2}{25} a^{11} + \frac{1}{25} a^{10} + \frac{7}{25} a^{9} - \frac{9}{25} a^{8} - \frac{11}{25} a^{7} - \frac{11}{25} a^{6} + \frac{3}{25} a^{5} + \frac{3}{25} a^{4} + \frac{34}{125} a^{3} + \frac{12}{25} a^{2} - \frac{11}{25} a + \frac{1}{5}$, $\frac{1}{625} a^{24} + \frac{1}{625} a^{23} + \frac{2}{625} a^{22} - \frac{2}{625} a^{21} + \frac{9}{625} a^{20} - \frac{3}{625} a^{19} - \frac{4}{625} a^{18} + \frac{8}{625} a^{17} + \frac{19}{625} a^{16} + \frac{62}{625} a^{15} + \frac{1}{625} a^{14} + \frac{3}{625} a^{13} + \frac{59}{625} a^{12} - \frac{3}{625} a^{11} + \frac{26}{625} a^{10} + \frac{263}{625} a^{9} - \frac{146}{625} a^{8} - \frac{148}{625} a^{7} - \frac{199}{625} a^{6} - \frac{307}{625} a^{5} - \frac{162}{625} a^{4} - \frac{84}{625} a^{3} + \frac{24}{625} a^{2} + \frac{17}{125} a - \frac{51}{125}$, $\frac{1}{625} a^{25} + \frac{1}{625} a^{23} + \frac{1}{625} a^{22} + \frac{1}{625} a^{21} + \frac{8}{625} a^{20} + \frac{9}{625} a^{19} - \frac{8}{625} a^{18} + \frac{1}{625} a^{17} - \frac{37}{625} a^{16} - \frac{1}{625} a^{15} + \frac{32}{625} a^{14} - \frac{29}{625} a^{13} + \frac{58}{625} a^{12} + \frac{39}{625} a^{11} - \frac{33}{625} a^{10} - \frac{119}{625} a^{9} + \frac{43}{625} a^{8} - \frac{291}{625} a^{7} - \frac{103}{625} a^{6} + \frac{32}{125} a^{5} - \frac{277}{625} a^{4} + \frac{18}{625} a^{3} + \frac{186}{625} a^{2} - \frac{58}{125} a + \frac{1}{125}$, $\frac{1}{203125} a^{26} + \frac{89}{203125} a^{25} + \frac{86}{203125} a^{24} - \frac{93}{40625} a^{23} + \frac{114}{40625} a^{22} + \frac{392}{203125} a^{21} - \frac{188}{15625} a^{20} - \frac{802}{203125} a^{19} - \frac{896}{203125} a^{18} - \frac{3228}{203125} a^{17} + \frac{2341}{203125} a^{16} - \frac{7652}{203125} a^{15} + \frac{13984}{203125} a^{14} - \frac{11328}{203125} a^{13} + \frac{17261}{203125} a^{12} - \frac{3282}{203125} a^{11} + \frac{4634}{203125} a^{10} + \frac{43847}{203125} a^{9} + \frac{30296}{203125} a^{8} - \frac{56997}{203125} a^{7} + \frac{96208}{203125} a^{6} + \frac{36108}{203125} a^{5} - \frac{12712}{40625} a^{4} - \frac{5877}{203125} a^{3} + \frac{899}{203125} a^{2} + \frac{14039}{40625} a - \frac{20076}{40625}$, $\frac{1}{3859375} a^{27} - \frac{1}{771875} a^{26} - \frac{161}{771875} a^{25} - \frac{99}{3859375} a^{24} + \frac{1316}{771875} a^{23} - \frac{6063}{3859375} a^{22} - \frac{11342}{3859375} a^{21} - \frac{26841}{3859375} a^{20} + \frac{14042}{3859375} a^{19} + \frac{29646}{3859375} a^{18} + \frac{2171}{296875} a^{17} - \frac{4106}{3859375} a^{16} - \frac{55178}{3859375} a^{15} + \frac{296576}{3859375} a^{14} + \frac{74918}{3859375} a^{13} + \frac{13436}{203125} a^{12} + \frac{119442}{3859375} a^{11} - \frac{352099}{3859375} a^{10} + \frac{740128}{3859375} a^{9} + \frac{576904}{3859375} a^{8} - \frac{1844274}{3859375} a^{7} + \frac{1266456}{3859375} a^{6} + \frac{186513}{3859375} a^{5} + \frac{918913}{3859375} a^{4} + \frac{1830587}{3859375} a^{3} + \frac{306464}{3859375} a^{2} + \frac{154608}{771875} a - \frac{198706}{771875}$, $\frac{1}{791171875} a^{28} - \frac{33}{791171875} a^{27} + \frac{1}{41640625} a^{26} - \frac{38716}{60859375} a^{25} - \frac{592549}{791171875} a^{24} - \frac{2990713}{791171875} a^{23} - \frac{61817}{41640625} a^{22} + \frac{1880313}{791171875} a^{21} - \frac{10712006}{791171875} a^{20} - \frac{13286798}{791171875} a^{19} + \frac{919421}{791171875} a^{18} + \frac{9928148}{791171875} a^{17} - \frac{49332116}{791171875} a^{16} + \frac{34528267}{791171875} a^{15} + \frac{61807621}{791171875} a^{14} + \frac{30571428}{791171875} a^{13} + \frac{78334614}{791171875} a^{12} + \frac{47447687}{791171875} a^{11} - \frac{62072794}{791171875} a^{10} - \frac{340685382}{791171875} a^{9} + \frac{11097556}{60859375} a^{8} - \frac{9398004}{158234375} a^{7} + \frac{292926292}{791171875} a^{6} + \frac{95383621}{791171875} a^{5} - \frac{244386567}{791171875} a^{4} - \frac{11045838}{158234375} a^{3} - \frac{367985936}{791171875} a^{2} + \frac{26600996}{158234375} a + \frac{66993884}{158234375}$, $\frac{1}{57259978785345951990064956178294756727441390455762359765625} a^{29} - \frac{80115082191993672434677743420801242893952913592}{131029699737633757414336284160857566882016911798083203125} a^{28} + \frac{3545454018219313329842273220045124641575987410630487}{57259978785345951990064956178294756727441390455762359765625} a^{27} + \frac{44418692934010721736115738797487892721787653661823793}{57259978785345951990064956178294756727441390455762359765625} a^{26} - \frac{11702776763330757271355061721533117635234766879767804231}{57259978785345951990064956178294756727441390455762359765625} a^{25} + \frac{484762348382722049502489334422046737000850930605892918}{1547566994198539242974728545359317749390307850155739453125} a^{24} - \frac{485374774150966163025071868807874604175218181934135403}{176184550108756775354046019010137713007511970633114953125} a^{23} - \frac{13561186307411110084006725433189084796795445000228110604}{57259978785345951990064956178294756727441390455762359765625} a^{22} + \frac{92016325723319092255124374482982869201099592329510710396}{57259978785345951990064956178294756727441390455762359765625} a^{21} + \frac{204185439599698401400382888491337990384883303092722699628}{57259978785345951990064956178294756727441390455762359765625} a^{20} - \frac{749456042040183927782998425991560790420702042865713228171}{57259978785345951990064956178294756727441390455762359765625} a^{19} - \frac{910005997944912542605553173708469039669484352618138402993}{57259978785345951990064956178294756727441390455762359765625} a^{18} + \frac{721482447662734982656827723452023960306028583906567757751}{57259978785345951990064956178294756727441390455762359765625} a^{17} - \frac{875377872253729290014460291351490902193893180315767035372}{57259978785345951990064956178294756727441390455762359765625} a^{16} + \frac{2868651057930485588844240113542065496503644521643405806789}{57259978785345951990064956178294756727441390455762359765625} a^{15} - \frac{4238232311724392056961763348796558165689642374421297939788}{57259978785345951990064956178294756727441390455762359765625} a^{14} + \frac{412668529364000783170113979759951509430219082232219027652}{4404613752718919383851150475253442825187799265827873828125} a^{13} + \frac{220684999787057837952461979192260581363850274917979797341}{2489564295015041390872389399056293770758321324163580859375} a^{12} - \frac{3976699355648752089019154432311165256029863973056942583446}{57259978785345951990064956178294756727441390455762359765625} a^{11} + \frac{613860815274150855113218584801694674339511652092850419367}{57259978785345951990064956178294756727441390455762359765625} a^{10} + \frac{1115923054709445192904613170000211074553057525772808543009}{2290399151413838079602598247131790269097655618230494390625} a^{9} + \frac{24472857766755547619227691255204310036789122849105720697642}{57259978785345951990064956178294756727441390455762359765625} a^{8} - \frac{8225249599852849617415291783673913864040817913153757010163}{57259978785345951990064956178294756727441390455762359765625} a^{7} + \frac{15842148329481239157224043322895841113168640642087934327439}{57259978785345951990064956178294756727441390455762359765625} a^{6} - \frac{438831921129891330379549567883869741777430278479891337959}{1547566994198539242974728545359317749390307850155739453125} a^{5} - \frac{5091898155205167340620157540508100063820016278042280127633}{57259978785345951990064956178294756727441390455762359765625} a^{4} + \frac{13357396879794371338922515493839876655694510269474362196754}{57259978785345951990064956178294756727441390455762359765625} a^{3} - \frac{203281118467605733101791632295623700326329162285696137606}{3013683093965576420529734535699724038286388971355913671875} a^{2} + \frac{670744663657965663248710871896476523315660418759205672193}{11451995757069190398012991235658951345488278091152471953125} a + \frac{90086415987944063885167643282370409613310832844127225028}{309513398839707848594945709071863549878061570031147890625}$

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

Class group and class number

$C_{5}$, which has order $5$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 9816173974474.957 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{14}\cdot 9816173974474.957 \cdot 5}{2\sqrt{11952412704796787125889897969349006221954345703125}}\approx 4.24358214727683$ (assuming GRH)

Galois group

$D_{30}$ (as 30T14):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.2855.1, 5.1.8151025.1, 6.2.40755125.1, 10.2.332196042753125.1, 15.1.1546118540397002491484375.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 sibling: Deg 30

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $30$ $30$ R $30$ $15^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{15}$ $30$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{3}$ $15^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
571Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.571.2t1.a.a$1$ $ 571 $ \(\Q(\sqrt{-571}) \) $C_2$ (as 2T1) $1$ $-1$
1.2855.2t1.a.a$1$ $ 5 \cdot 571 $ \(\Q(\sqrt{-2855}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.2855.6t3.a.a$2$ $ 5 \cdot 571 $ 6.0.4654235275.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.2855.3t2.a.a$2$ $ 5 \cdot 571 $ 3.1.2855.1 $S_3$ (as 3T2) $1$ $0$
* 2.2855.5t2.a.a$2$ $ 5 \cdot 571 $ 5.1.8151025.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2855.5t2.a.b$2$ $ 5 \cdot 571 $ 5.1.8151025.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2855.10t3.a.a$2$ $ 5 \cdot 571 $ 10.0.37936788082406875.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.2855.10t3.a.b$2$ $ 5 \cdot 571 $ 10.0.37936788082406875.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.2855.15t2.a.b$2$ $ 5 \cdot 571 $ 15.1.1546118540397002491484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2855.30t14.a.b$2$ $ 5 \cdot 571 $ 30.2.11952412704796787125889897969349006221954345703125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2855.15t2.a.d$2$ $ 5 \cdot 571 $ 15.1.1546118540397002491484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2855.30t14.a.d$2$ $ 5 \cdot 571 $ 30.2.11952412704796787125889897969349006221954345703125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2855.15t2.a.c$2$ $ 5 \cdot 571 $ 15.1.1546118540397002491484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2855.30t14.a.a$2$ $ 5 \cdot 571 $ 30.2.11952412704796787125889897969349006221954345703125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2855.15t2.a.a$2$ $ 5 \cdot 571 $ 15.1.1546118540397002491484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2855.30t14.a.c$2$ $ 5 \cdot 571 $ 30.2.11952412704796787125889897969349006221954345703125.1 $D_{30}$ (as 30T14) $1$ $0$

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 *.