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Properties

Label 30.2.605...125.1
Degree $30$
Signature $[2, 14]$
Discriminant $6.055\times 10^{43}$
Root discriminant $28.80$
Ramified primes $5, 239$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 4*x^29 - 4*x^28 + 32*x^27 + 29*x^26 - 207*x^25 - 80*x^24 + 808*x^23 + 324*x^22 - 2807*x^21 - 784*x^20 + 6744*x^19 + 2549*x^18 - 11743*x^17 - 5231*x^16 + 15441*x^15 + 10445*x^14 - 10233*x^13 - 7776*x^12 + 6706*x^11 + 5598*x^10 - 2410*x^9 - 2278*x^8 + 1045*x^7 + 940*x^6 - 185*x^5 - 176*x^4 + 44*x^3 + 30*x^2 - 6*x - 1)
 
gp: K = bnfinit(x^30 - 4*x^29 - 4*x^28 + 32*x^27 + 29*x^26 - 207*x^25 - 80*x^24 + 808*x^23 + 324*x^22 - 2807*x^21 - 784*x^20 + 6744*x^19 + 2549*x^18 - 11743*x^17 - 5231*x^16 + 15441*x^15 + 10445*x^14 - 10233*x^13 - 7776*x^12 + 6706*x^11 + 5598*x^10 - 2410*x^9 - 2278*x^8 + 1045*x^7 + 940*x^6 - 185*x^5 - 176*x^4 + 44*x^3 + 30*x^2 - 6*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -6, 30, 44, -176, -185, 940, 1045, -2278, -2410, 5598, 6706, -7776, -10233, 10445, 15441, -5231, -11743, 2549, 6744, -784, -2807, 324, 808, -80, -207, 29, 32, -4, -4, 1]);
 

\(x^{30} - 4 x^{29} - 4 x^{28} + 32 x^{27} + 29 x^{26} - 207 x^{25} - 80 x^{24} + 808 x^{23} + 324 x^{22} - 2807 x^{21} - 784 x^{20} + 6744 x^{19} + 2549 x^{18} - 11743 x^{17} - 5231 x^{16} + 15441 x^{15} + 10445 x^{14} - 10233 x^{13} - 7776 x^{12} + 6706 x^{11} + 5598 x^{10} - 2410 x^{9} - 2278 x^{8} + 1045 x^{7} + 940 x^{6} - 185 x^{5} - 176 x^{4} + 44 x^{3} + 30 x^{2} - 6 x - 1\)  Toggle raw display

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:  \(60550910545856385116788597847358428955078125\)\(\medspace = 5^{15}\cdot 239^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $28.80$
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:  $5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{7} a^{24} - \frac{2}{7} a^{23} + \frac{3}{7} a^{22} + \frac{3}{7} a^{21} + \frac{2}{7} a^{20} - \frac{3}{7} a^{19} - \frac{2}{7} a^{18} + \frac{1}{7} a^{17} - \frac{3}{7} a^{16} - \frac{3}{7} a^{15} - \frac{2}{7} a^{14} - \frac{1}{7} a^{13} - \frac{3}{7} a^{12} - \frac{1}{7} a^{11} + \frac{2}{7} a^{9} - \frac{3}{7} a^{8} + \frac{1}{7} a^{6} + \frac{3}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{25} - \frac{1}{7} a^{23} + \frac{2}{7} a^{22} + \frac{1}{7} a^{21} + \frac{1}{7} a^{20} - \frac{1}{7} a^{19} - \frac{3}{7} a^{18} - \frac{1}{7} a^{17} - \frac{2}{7} a^{16} - \frac{1}{7} a^{15} + \frac{2}{7} a^{14} + \frac{2}{7} a^{13} - \frac{2}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{91} a^{26} + \frac{4}{91} a^{25} - \frac{5}{91} a^{24} + \frac{1}{7} a^{23} - \frac{17}{91} a^{22} + \frac{1}{13} a^{21} + \frac{37}{91} a^{20} + \frac{40}{91} a^{19} + \frac{16}{91} a^{18} - \frac{17}{91} a^{17} - \frac{3}{7} a^{16} + \frac{38}{91} a^{15} + \frac{32}{91} a^{14} - \frac{37}{91} a^{13} + \frac{24}{91} a^{12} + \frac{5}{91} a^{11} + \frac{30}{91} a^{10} + \frac{32}{91} a^{9} + \frac{24}{91} a^{8} - \frac{40}{91} a^{7} - \frac{1}{13} a^{6} - \frac{1}{13} a^{5} - \frac{25}{91} a^{4} + \frac{22}{91} a^{3} + \frac{2}{13} a^{2} + \frac{19}{91} a - \frac{10}{91}$, $\frac{1}{91} a^{27} + \frac{5}{91} a^{25} - \frac{6}{91} a^{24} - \frac{17}{91} a^{23} + \frac{10}{91} a^{22} + \frac{9}{91} a^{21} + \frac{22}{91} a^{20} + \frac{38}{91} a^{19} + \frac{10}{91} a^{18} - \frac{36}{91} a^{17} - \frac{2}{13} a^{16} - \frac{29}{91} a^{15} - \frac{5}{13} a^{14} - \frac{10}{91} a^{13} + \frac{2}{7} a^{12} - \frac{3}{91} a^{11} - \frac{36}{91} a^{10} + \frac{2}{7} a^{9} + \frac{1}{13} a^{8} - \frac{3}{91} a^{7} + \frac{3}{13} a^{6} + \frac{16}{91} a^{5} + \frac{44}{91} a^{4} - \frac{5}{13} a^{3} + \frac{15}{91} a^{2} + \frac{5}{91} a - \frac{38}{91}$, $\frac{1}{14676773839} a^{28} + \frac{3707215}{14676773839} a^{27} - \frac{6303146}{2096681977} a^{26} - \frac{981496800}{14676773839} a^{25} + \frac{555778896}{14676773839} a^{24} + \frac{6249378762}{14676773839} a^{23} + \frac{1781905890}{14676773839} a^{22} - \frac{3354810841}{14676773839} a^{21} - \frac{313809588}{772461781} a^{20} + \frac{5130237142}{14676773839} a^{19} + \frac{2122701111}{14676773839} a^{18} + \frac{5849101380}{14676773839} a^{17} - \frac{6851951881}{14676773839} a^{16} + \frac{5459439599}{14676773839} a^{15} + \frac{4499370672}{14676773839} a^{14} + \frac{377439652}{1128982603} a^{13} + \frac{4334058208}{14676773839} a^{12} - \frac{13751700}{84836843} a^{11} + \frac{50944304}{110351683} a^{10} + \frac{1029226629}{2096681977} a^{9} - \frac{579245126}{14676773839} a^{8} + \frac{2131576129}{14676773839} a^{7} + \frac{3122865965}{14676773839} a^{6} + \frac{5744248316}{14676773839} a^{5} + \frac{834951158}{14676773839} a^{4} - \frac{409863199}{1128982603} a^{3} + \frac{6701117463}{14676773839} a^{2} - \frac{2631267697}{14676773839} a - \frac{6162129166}{14676773839}$, $\frac{1}{5326004272701994741133641125737739123181} a^{29} - \frac{21100556403623109705330858439}{5326004272701994741133641125737739123181} a^{28} + \frac{163401106993758781968293063828526751}{33080771880136613298966715066694031821} a^{27} - \frac{26152778473005008545220673223843016808}{5326004272701994741133641125737739123181} a^{26} - \frac{77435642835076243374634294616616270689}{5326004272701994741133641125737739123181} a^{25} - \frac{344549216750484713945858659693140443585}{5326004272701994741133641125737739123181} a^{24} + \frac{1205423290308603668481270779601834116814}{5326004272701994741133641125737739123181} a^{23} - \frac{2654010275795164745486181288809238691016}{5326004272701994741133641125737739123181} a^{22} - \frac{273144677527698097100755949383043684163}{760857753243142105876234446533962731883} a^{21} - \frac{603921414483900151882515712298086429497}{5326004272701994741133641125737739123181} a^{20} + \frac{6029008943849748856391879935878526993}{409692636361691903164126240441364547937} a^{19} + \frac{1930815325331553740873696942985825068786}{5326004272701994741133641125737739123181} a^{18} + \frac{651092922992269447579773757936042335256}{5326004272701994741133641125737739123181} a^{17} + \frac{1525005027811712548561237095259968561133}{5326004272701994741133641125737739123181} a^{16} - \frac{1848765118466910092793325891156547721648}{5326004272701994741133641125737739123181} a^{15} - \frac{51614849592237624252288018398566417776}{5326004272701994741133641125737739123181} a^{14} + \frac{995883015125920160740347168455393108026}{5326004272701994741133641125737739123181} a^{13} + \frac{2201756860991824293017811156334268056830}{5326004272701994741133641125737739123181} a^{12} + \frac{85450538084609085705406503582973650945}{409692636361691903164126240441364547937} a^{11} + \frac{143664359107704026888665141670645463429}{760857753243142105876234446533962731883} a^{10} + \frac{1134710825240496170850810996007580056803}{5326004272701994741133641125737739123181} a^{9} + \frac{694461040099171528415900634320762156669}{5326004272701994741133641125737739123181} a^{8} + \frac{1574720138595661006722107605254823174450}{5326004272701994741133641125737739123181} a^{7} - \frac{9514221983782318645519655055625031446}{40045144907533795046117602449155933257} a^{6} - \frac{2270856784560219610689362836788719342240}{5326004272701994741133641125737739123181} a^{5} + \frac{1269044698310704865126226151433776627600}{5326004272701994741133641125737739123181} a^{4} - \frac{1304258640655892062904837203073106596912}{5326004272701994741133641125737739123181} a^{3} - \frac{2058240917031049936995578313354674432747}{5326004272701994741133641125737739123181} a^{2} + \frac{2561441189281033997854680796076099350644}{5326004272701994741133641125737739123181} a - \frac{1535409928942451191968477654787877446630}{5326004272701994741133641125737739123181}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (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 $2$)  Toggle raw display
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:  \( 5581738694.683278 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{14}\cdot 5581738694.683278 \cdot 1}{2\sqrt{60550910545856385116788597847358428955078125}}\approx 0.214416061763025$ (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

\(\Q(\sqrt{5}) \), 3.1.239.1, 5.1.57121.1, 6.2.7140125.1, 10.2.10196277003125.1, 15.1.44543599279432079.1

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 $30$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{3}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{15}$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
239Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1195.2t1.a.a$1$ $ 5 \cdot 239 $ \(\Q(\sqrt{-1195}) \) $C_2$ (as 2T1) $1$ $-1$
1.239.2t1.a.a$1$ $ 239 $ \(\Q(\sqrt{-239}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.5975.6t3.a.a$2$ $ 5^{2} \cdot 239 $ 6.0.1706489875.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.239.3t2.a.a$2$ $ 239 $ 3.1.239.1 $S_3$ (as 3T2) $1$ $0$
* 2.239.5t2.a.b$2$ $ 239 $ 5.1.57121.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.239.5t2.a.a$2$ $ 239 $ 5.1.57121.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.5975.10t3.a.b$2$ $ 5^{2} \cdot 239 $ 10.0.2436910203746875.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.5975.10t3.a.a$2$ $ 5^{2} \cdot 239 $ 10.0.2436910203746875.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.239.15t2.a.c$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.5975.30t14.a.a$2$ $ 5^{2} \cdot 239 $ 30.2.60550910545856385116788597847358428955078125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.239.15t2.a.a$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.5975.30t14.a.c$2$ $ 5^{2} \cdot 239 $ 30.2.60550910545856385116788597847358428955078125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.239.15t2.a.d$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.5975.30t14.a.d$2$ $ 5^{2} \cdot 239 $ 30.2.60550910545856385116788597847358428955078125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.239.15t2.a.b$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.5975.30t14.a.b$2$ $ 5^{2} \cdot 239 $ 30.2.60550910545856385116788597847358428955078125.1 $D_{30}$ (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 *.