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Properties

Label 30.2.305...688.1
Degree $30$
Signature $[2, 14]$
Discriminant $3.057\times 10^{49}$
Root discriminant $44.62$
Ramified primes $2, 3, 239$
Class number not computed
Class group not computed
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 24*x^28 + 342*x^26 - 3240*x^24 + 23571*x^22 - 138753*x^20 + 675054*x^18 - 2755620*x^16 + 9454401*x^14 - 26985393*x^12 + 64658655*x^10 - 119042784*x^8 + 183878586*x^6 - 172186884*x^4 + 153055008*x^2 - 14348907)
 
gp: K = bnfinit(x^30 - 24*x^28 + 342*x^26 - 3240*x^24 + 23571*x^22 - 138753*x^20 + 675054*x^18 - 2755620*x^16 + 9454401*x^14 - 26985393*x^12 + 64658655*x^10 - 119042784*x^8 + 183878586*x^6 - 172186884*x^4 + 153055008*x^2 - 14348907, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14348907, 0, 153055008, 0, -172186884, 0, 183878586, 0, -119042784, 0, 64658655, 0, -26985393, 0, 9454401, 0, -2755620, 0, 675054, 0, -138753, 0, 23571, 0, -3240, 0, 342, 0, -24, 0, 1]);
 

\( x^{30} - 24 x^{28} + 342 x^{26} - 3240 x^{24} + 23571 x^{22} - 138753 x^{20} + 675054 x^{18} - 2755620 x^{16} + 9454401 x^{14} - 26985393 x^{12} + 64658655 x^{10} - 119042784 x^{8} + 183878586 x^{6} - 172186884 x^{4} + 153055008 x^{2} - 14348907 \)

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:  \(30569568178695653232311243684564905832093935730688\)\(\medspace = 2^{30}\cdot 3^{15}\cdot 239^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $44.62$
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:  $2, 3, 239$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{2187} a^{14}$, $\frac{1}{2187} a^{15}$, $\frac{1}{6561} a^{16}$, $\frac{1}{6561} a^{17}$, $\frac{1}{19683} a^{18}$, $\frac{1}{19683} a^{19}$, $\frac{1}{59049} a^{20}$, $\frac{1}{59049} a^{21}$, $\frac{1}{177147} a^{22}$, $\frac{1}{177147} a^{23}$, $\frac{1}{6908733} a^{24} - \frac{2}{767637} a^{22} - \frac{2}{85293} a^{18} - \frac{2}{85293} a^{16} - \frac{1}{9477} a^{14} + \frac{2}{9477} a^{12} - \frac{2}{3159} a^{10} + \frac{1}{351} a^{8} - \frac{2}{117} a^{6} - \frac{1}{39} a^{4} + \frac{2}{39} a^{2} + \frac{4}{13}$, $\frac{1}{6908733} a^{25} - \frac{2}{767637} a^{23} - \frac{2}{85293} a^{19} - \frac{2}{85293} a^{17} - \frac{1}{9477} a^{15} + \frac{2}{9477} a^{13} - \frac{2}{3159} a^{11} + \frac{1}{351} a^{9} - \frac{2}{117} a^{7} - \frac{1}{39} a^{5} + \frac{2}{39} a^{3} + \frac{4}{13} a$, $\frac{1}{20726199} a^{26} + \frac{1}{767637} a^{22} - \frac{2}{255879} a^{20} + \frac{1}{255879} a^{18} - \frac{2}{85293} a^{16} - \frac{1}{9477} a^{14} - \frac{1}{3159} a^{12} + \frac{4}{3159} a^{10} - \frac{1}{1053} a^{8} - \frac{1}{39} a^{4} + \frac{1}{13} a^{2} - \frac{2}{13}$, $\frac{1}{20726199} a^{27} + \frac{1}{767637} a^{23} - \frac{2}{255879} a^{21} + \frac{1}{255879} a^{19} - \frac{2}{85293} a^{17} - \frac{1}{9477} a^{15} - \frac{1}{3159} a^{13} + \frac{4}{3159} a^{11} - \frac{1}{1053} a^{9} - \frac{1}{39} a^{5} + \frac{1}{13} a^{3} - \frac{2}{13} a$, $\frac{1}{25161311875033971} a^{28} - \frac{8108245}{645161842949589} a^{26} - \frac{75424400}{2795701319448219} a^{24} + \frac{293572879}{310633479938691} a^{22} - \frac{1173121429}{310633479938691} a^{20} + \frac{107436752}{14792070473271} a^{18} - \frac{106724084}{34514831104299} a^{16} + \frac{590740931}{3834981233811} a^{14} + \frac{186200132}{294998556447} a^{12} - \frac{307366898}{182618153991} a^{10} + \frac{1189733240}{426109025979} a^{8} + \frac{1364538158}{142036341993} a^{6} - \frac{536270992}{15781815777} a^{4} + \frac{836226970}{15781815777} a^{2} + \frac{2115234932}{5260605259}$, $\frac{1}{25161311875033971} a^{29} - \frac{8108245}{645161842949589} a^{27} - \frac{75424400}{2795701319448219} a^{25} + \frac{293572879}{310633479938691} a^{23} - \frac{1173121429}{310633479938691} a^{21} + \frac{107436752}{14792070473271} a^{19} - \frac{106724084}{34514831104299} a^{17} + \frac{590740931}{3834981233811} a^{15} + \frac{186200132}{294998556447} a^{13} - \frac{307366898}{182618153991} a^{11} + \frac{1189733240}{426109025979} a^{9} + \frac{1364538158}{142036341993} a^{7} - \frac{536270992}{15781815777} a^{5} + \frac{836226970}{15781815777} a^{3} + \frac{2115234932}{5260605259} a$

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

Class group and class number

not computed

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:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

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

\(\Q(\sqrt{3}) \), 3.1.239.1, 5.1.57121.1, 6.2.98705088.2, 10.2.811891199757312.1, 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 R R $30$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $30$ $30$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
239Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2868.2t1.a.a$1$ $ 2^{2} \cdot 3 \cdot 239 $ \(\Q(\sqrt{-717}) \) $C_2$ (as 2T1) $1$ $-1$
1.239.2t1.a.a$1$ $ 239 $ \(\Q(\sqrt{-239}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.12.2t1.a.a$1$ $ 2^{2} \cdot 3 $ \(\Q(\sqrt{3}) \) $C_2$ (as 2T1) $1$ $1$
* 2.34416.6t3.a.a$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 6.0.23590516032.3 $D_{6}$ (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.a$2$ $ 239 $ 5.1.57121.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.239.5t2.a.b$2$ $ 239 $ 5.1.57121.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.34416.10t3.a.a$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 10.0.194041996741997568.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.34416.10t3.a.b$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 10.0.194041996741997568.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.239.15t2.a.b$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.239.15t2.a.a$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.239.15t2.a.d$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.239.15t2.a.c$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.34416.30t14.a.a$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 30.2.30569568178695653232311243684564905832093935730688.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.34416.30t14.a.d$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 30.2.30569568178695653232311243684564905832093935730688.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.34416.30t14.a.b$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 30.2.30569568178695653232311243684564905832093935730688.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.34416.30t14.a.c$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 30.2.30569568178695653232311243684564905832093935730688.1 $D_{30}$ (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 *.