Properties

Label 30.2.140...000.1
Degree $30$
Signature $[2, 14]$
Discriminant $1.403\times 10^{46}$
Root discriminant $34.53$
Ramified primes $2, 5, 131$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 11*x^28 + 4*x^27 + 44*x^26 - 6*x^25 - 74*x^24 - 446*x^23 - 1543*x^22 + 183*x^21 - 1721*x^20 + 7116*x^19 + 11884*x^18 - 3168*x^17 + 7664*x^16 - 18756*x^15 + 17109*x^14 - 3363*x^13 - 2431*x^12 + 11594*x^11 - 15080*x^10 + 6868*x^9 + 3884*x^8 - 3768*x^7 + 2512*x^6 - 1864*x^5 - 1696*x^4 + 656*x^3 + 448*x^2 + 16*x - 16)
 
gp: K = bnfinit(x^30 - x^29 + 11*x^28 + 4*x^27 + 44*x^26 - 6*x^25 - 74*x^24 - 446*x^23 - 1543*x^22 + 183*x^21 - 1721*x^20 + 7116*x^19 + 11884*x^18 - 3168*x^17 + 7664*x^16 - 18756*x^15 + 17109*x^14 - 3363*x^13 - 2431*x^12 + 11594*x^11 - 15080*x^10 + 6868*x^9 + 3884*x^8 - 3768*x^7 + 2512*x^6 - 1864*x^5 - 1696*x^4 + 656*x^3 + 448*x^2 + 16*x - 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, 16, 448, 656, -1696, -1864, 2512, -3768, 3884, 6868, -15080, 11594, -2431, -3363, 17109, -18756, 7664, -3168, 11884, 7116, -1721, 183, -1543, -446, -74, -6, 44, 4, 11, -1, 1]);
 

\( x^{30} - x^{29} + 11 x^{28} + 4 x^{27} + 44 x^{26} - 6 x^{25} - 74 x^{24} - 446 x^{23} - 1543 x^{22} + 183 x^{21} - 1721 x^{20} + 7116 x^{19} + 11884 x^{18} - 3168 x^{17} + 7664 x^{16} - 18756 x^{15} + 17109 x^{14} - 3363 x^{13} - 2431 x^{12} + 11594 x^{11} - 15080 x^{10} + 6868 x^{9} + 3884 x^{8} - 3768 x^{7} + 2512 x^{6} - 1864 x^{5} - 1696 x^{4} + 656 x^{3} + 448 x^{2} + 16 x - 16 \)

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:  \(14026461290181639205847118647072000000000000000\)\(\medspace = 2^{20}\cdot 5^{15}\cdot 131^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $34.53$
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:  $2, 5, 131$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{20} - \frac{1}{8} a^{19} - \frac{1}{8} a^{17} - \frac{1}{8} a^{16} - \frac{1}{8} a^{14} + \frac{1}{8} a^{13} + \frac{1}{8} a^{12} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{2} a^{8} + \frac{1}{8} a^{7} + \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{21} - \frac{1}{8} a^{19} - \frac{1}{8} a^{18} - \frac{1}{8} a^{16} - \frac{1}{8} a^{15} - \frac{1}{4} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} + \frac{3}{8} a^{8} - \frac{1}{2} a^{7} - \frac{3}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{8} a^{22} - \frac{1}{8} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{23} - \frac{1}{8} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{24} - \frac{1}{8} a^{16} - \frac{3}{8} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{25} - \frac{1}{8} a^{17} + \frac{1}{8} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{181424} a^{26} + \frac{5153}{181424} a^{25} - \frac{10609}{181424} a^{24} + \frac{400}{11339} a^{23} + \frac{45}{1564} a^{22} - \frac{929}{90712} a^{21} - \frac{89}{2668} a^{20} + \frac{2355}{45356} a^{19} - \frac{6887}{181424} a^{18} - \frac{10173}{181424} a^{17} - \frac{7519}{181424} a^{16} + \frac{395}{45356} a^{15} + \frac{9661}{90712} a^{14} - \frac{19939}{90712} a^{13} + \frac{213}{3128} a^{12} - \frac{11233}{45356} a^{11} + \frac{39543}{181424} a^{10} - \frac{41403}{181424} a^{9} - \frac{8767}{181424} a^{8} - \frac{22453}{45356} a^{7} - \frac{15647}{90712} a^{6} - \frac{10265}{45356} a^{5} - \frac{3879}{22678} a^{4} + \frac{4739}{45356} a^{3} - \frac{7839}{45356} a^{2} - \frac{10335}{22678} a + \frac{5421}{22678}$, $\frac{1}{362848} a^{27} - \frac{1}{362848} a^{26} - \frac{13233}{362848} a^{25} - \frac{7075}{181424} a^{24} + \frac{8055}{181424} a^{23} + \frac{1631}{45356} a^{22} + \frac{11321}{181424} a^{21} + \frac{9589}{181424} a^{20} + \frac{18709}{362848} a^{19} - \frac{5645}{362848} a^{18} - \frac{7413}{362848} a^{17} - \frac{12437}{181424} a^{16} - \frac{2637}{181424} a^{15} + \frac{471}{7888} a^{14} - \frac{239}{2668} a^{13} - \frac{20745}{90712} a^{12} + \frac{8657}{362848} a^{11} - \frac{38839}{362848} a^{10} + \frac{4993}{362848} a^{9} + \frac{4265}{45356} a^{8} - \frac{67147}{181424} a^{7} + \frac{659}{7888} a^{6} + \frac{24295}{90712} a^{5} + \frac{8157}{90712} a^{4} - \frac{1073}{3128} a^{3} + \frac{1837}{11339} a^{2} + \frac{1389}{45356} a - \frac{11877}{45356}$, $\frac{1}{362848} a^{28} - \frac{2649}{362848} a^{25} - \frac{10363}{181424} a^{24} + \frac{875}{181424} a^{23} - \frac{2687}{181424} a^{22} - \frac{2339}{90712} a^{21} - \frac{941}{362848} a^{20} + \frac{2441}{22678} a^{19} - \frac{8853}{90712} a^{18} - \frac{45161}{362848} a^{17} + \frac{431}{7888} a^{16} + \frac{4249}{90712} a^{15} + \frac{12369}{181424} a^{14} + \frac{9387}{90712} a^{13} - \frac{44511}{362848} a^{12} + \frac{1123}{181424} a^{11} + \frac{1511}{90712} a^{10} - \frac{1945}{21344} a^{9} - \frac{25501}{90712} a^{8} - \frac{978}{11339} a^{7} - \frac{49223}{181424} a^{6} - \frac{9347}{45356} a^{5} + \frac{212}{493} a^{4} + \frac{17513}{90712} a^{3} - \frac{10009}{22678} a^{2} - \frac{3215}{11339} a + \frac{21801}{45356}$, $\frac{1}{69207240148595255768335588845025439843535311783488} a^{29} - \frac{73180399435857936923844912527478022741978369}{69207240148595255768335588845025439843535311783488} a^{28} - \frac{44426420325075816096022822366042352470374501}{69207240148595255768335588845025439843535311783488} a^{27} - \frac{14674936867960527898946857923432544401704097}{34603620074297627884167794422512719921767655891744} a^{26} - \frac{368350032857284609706886263439348744733813924285}{34603620074297627884167794422512719921767655891744} a^{25} + \frac{54672872888706054332304859779963008288765360951}{1017753531596989055416699835956256468287283996816} a^{24} - \frac{1059600419127350014874743523339479537084730145873}{34603620074297627884167794422512719921767655891744} a^{23} - \frac{2102412659978950673833992246167014834516710026061}{34603620074297627884167794422512719921767655891744} a^{22} - \frac{1814132652249714625196320959593061851900240436927}{69207240148595255768335588845025439843535311783488} a^{21} - \frac{60170964477567407377533141222800581795597539005}{4071014126387956221666799343825025873149135987264} a^{20} - \frac{8623987878833241703204021584146008068535273442797}{69207240148595255768335588845025439843535311783488} a^{19} - \frac{1606658336569914886159611008377117707736349837797}{34603620074297627884167794422512719921767655891744} a^{18} - \frac{50666343641588027792700745486492730850416412041}{474022192798597642248873896198804382489967888928} a^{17} - \frac{4111643136666781662885944859125925599296830160755}{34603620074297627884167794422512719921767655891744} a^{16} + \frac{99176417471279577918728453565355425998557756028}{1081363127321800871380243575703522497555239246617} a^{15} + \frac{117894924608295337091941616259772013255650147321}{4325452509287203485520974302814089990220956986468} a^{14} + \frac{10052808969510518543915378696970754221178114063265}{69207240148595255768335588845025439843535311783488} a^{13} + \frac{928276871250240533137256097366922266029059869229}{4071014126387956221666799343825025873149135987264} a^{12} + \frac{7172924475322103235690411322036868489689517035417}{69207240148595255768335588845025439843535311783488} a^{11} - \frac{4277433338784604307961672459872702981266751219895}{17301810037148813942083897211256359960883827945872} a^{10} + \frac{8381796099988628299085040376168523714964243441}{99722248052730916092702577586491988247169037152} a^{9} + \frac{3699552078537770017195817735975466660275127476313}{34603620074297627884167794422512719921767655891744} a^{8} - \frac{1517620159607030291823422337684870014359156305503}{8650905018574406971041948605628179980441913972936} a^{7} + \frac{1564537916546240308847508455564030875090848258603}{17301810037148813942083897211256359960883827945872} a^{6} - \frac{95680421483383365592652271445381379068050804617}{237011096399298821124436948099402191244983944464} a^{5} + \frac{1994481174780572043362373790168445740744389825119}{4325452509287203485520974302814089990220956986468} a^{4} - \frac{240413742370011534608473297470344080717828193015}{8650905018574406971041948605628179980441913972936} a^{3} + \frac{128121617103472064974706247822281626878500279455}{298307069606014033484205124332006206222134964584} a^{2} - \frac{1161271677517767613850463386915938783259769142819}{4325452509287203485520974302814089990220956986468} a + \frac{131745549404783755027845395370497925128033012698}{1081363127321800871380243575703522497555239246617}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 145673548066.95038 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{14}\cdot 145673548066.95038 \cdot 2}{2\sqrt{14026461290181639205847118647072000000000000000}}\approx 0.735334238100197$ (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

\(\Q(\sqrt{5}) \), 3.1.524.1, 5.1.17161.1, 6.2.34322000.3, 10.2.920312253125.1, 15.1.677952124826430464.1

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 R $30$ R $30$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{6}$ $30$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{15}$ $15^{2}$ $30$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{5}$ $15^{2}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$131$$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.655.2t1.a.a$1$ $ 5 \cdot 131 $ \(\Q(\sqrt{-655}) \) $C_2$ (as 2T1) $1$ $-1$
1.131.2t1.a.a$1$ $ 131 $ \(\Q(\sqrt{-131}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.13100.6t3.b.a$2$ $ 2^{2} \cdot 5^{2} \cdot 131 $ 6.0.4496182000.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.524.3t2.a.a$2$ $ 2^{2} \cdot 131 $ 3.1.524.1 $S_3$ (as 3T2) $1$ $0$
* 2.131.5t2.a.b$2$ $ 131 $ 5.1.17161.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.131.5t2.a.a$2$ $ 131 $ 5.1.17161.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.3275.10t3.a.b$2$ $ 5^{2} \cdot 131 $ 10.0.120560905159375.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.3275.10t3.a.a$2$ $ 5^{2} \cdot 131 $ 10.0.120560905159375.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.524.15t2.a.a$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.13100.30t14.a.c$2$ $ 2^{2} \cdot 5^{2} \cdot 131 $ 30.2.14026461290181639205847118647072000000000000000.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.524.15t2.a.c$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.13100.30t14.a.a$2$ $ 2^{2} \cdot 5^{2} \cdot 131 $ 30.2.14026461290181639205847118647072000000000000000.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.524.15t2.a.b$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.13100.30t14.a.b$2$ $ 2^{2} \cdot 5^{2} \cdot 131 $ 30.2.14026461290181639205847118647072000000000000000.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.524.15t2.a.d$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.13100.30t14.a.d$2$ $ 2^{2} \cdot 5^{2} \cdot 131 $ 30.2.14026461290181639205847118647072000000000000000.1 $D_{30}$ (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 *.