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Properties

Label 30.2.698...512.1
Degree $30$
Signature $[2, 14]$
Discriminant $6.981\times 10^{46}$
Root discriminant $36.43$
Ramified primes $2, 239$
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 - 16*x^28 + 152*x^26 - 960*x^24 + 4656*x^22 - 18272*x^20 + 59264*x^18 - 161280*x^16 + 368896*x^14 - 701952*x^12 + 1121280*x^10 - 1376256*x^8 + 1417216*x^6 - 884736*x^4 + 524288*x^2 - 32768)
 
gp: K = bnfinit(x^30 - 16*x^28 + 152*x^26 - 960*x^24 + 4656*x^22 - 18272*x^20 + 59264*x^18 - 161280*x^16 + 368896*x^14 - 701952*x^12 + 1121280*x^10 - 1376256*x^8 + 1417216*x^6 - 884736*x^4 + 524288*x^2 - 32768, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32768, 0, 524288, 0, -884736, 0, 1417216, 0, -1376256, 0, 1121280, 0, -701952, 0, 368896, 0, -161280, 0, 59264, 0, -18272, 0, 4656, 0, -960, 0, 152, 0, -16, 0, 1]);
 

\( x^{30} - 16 x^{28} + 152 x^{26} - 960 x^{24} + 4656 x^{22} - 18272 x^{20} + 59264 x^{18} - 161280 x^{16} + 368896 x^{14} - 701952 x^{12} + 1121280 x^{10} - 1376256 x^{8} + 1417216 x^{6} - 884736 x^{4} + 524288 x^{2} - 32768 \)

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:  \(69810446891843341455650582518642209772915392512\)\(\medspace = 2^{45}\cdot 239^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $36.43$
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:  $2, 239$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{53248} a^{24} - \frac{3}{13312} a^{22} - \frac{3}{3328} a^{18} - \frac{1}{1664} a^{16} - \frac{3}{1664} a^{14} + \frac{1}{416} a^{12} - \frac{1}{208} a^{10} + \frac{3}{208} a^{8} - \frac{3}{52} a^{6} - \frac{3}{52} a^{4} + \frac{1}{13} a^{2} + \frac{4}{13}$, $\frac{1}{53248} a^{25} - \frac{3}{13312} a^{23} - \frac{3}{3328} a^{19} - \frac{1}{1664} a^{17} - \frac{3}{1664} a^{15} + \frac{1}{416} a^{13} - \frac{1}{208} a^{11} + \frac{3}{208} a^{9} - \frac{3}{52} a^{7} - \frac{3}{52} a^{5} + \frac{1}{13} a^{3} + \frac{4}{13} a$, $\frac{1}{106496} a^{26} + \frac{3}{26624} a^{22} - \frac{3}{6656} a^{20} + \frac{1}{6656} a^{18} - \frac{1}{1664} a^{16} - \frac{3}{1664} a^{14} - \frac{3}{832} a^{12} + \frac{1}{104} a^{10} - \frac{1}{208} a^{8} - \frac{3}{52} a^{4} + \frac{3}{26} a^{2} - \frac{2}{13}$, $\frac{1}{106496} a^{27} + \frac{3}{26624} a^{23} - \frac{3}{6656} a^{21} + \frac{1}{6656} a^{19} - \frac{1}{1664} a^{17} - \frac{3}{1664} a^{15} - \frac{3}{832} a^{13} + \frac{1}{104} a^{11} - \frac{1}{208} a^{9} - \frac{3}{52} a^{5} + \frac{3}{26} a^{3} - \frac{2}{13} a$, $\frac{1}{86189756563456} a^{28} - \frac{8108245}{3314990637056} a^{26} - \frac{4714025}{1346714946304} a^{24} + \frac{880718637}{10773719570432} a^{22} - \frac{1173121429}{5386859785216} a^{20} + \frac{6714797}{24048481184} a^{18} - \frac{26681021}{336678736576} a^{16} + \frac{1772222793}{673357473152} a^{14} + \frac{46550033}{6474591088} a^{12} - \frac{153683449}{12024240592} a^{10} + \frac{148716655}{10521210518} a^{8} + \frac{682269079}{21042421036} a^{6} - \frac{402203244}{5260605259} a^{4} + \frac{418113485}{5260605259} a^{2} + \frac{2115234932}{5260605259}$, $\frac{1}{86189756563456} a^{29} - \frac{8108245}{3314990637056} a^{27} - \frac{4714025}{1346714946304} a^{25} + \frac{880718637}{10773719570432} a^{23} - \frac{1173121429}{5386859785216} a^{21} + \frac{6714797}{24048481184} a^{19} - \frac{26681021}{336678736576} a^{17} + \frac{1772222793}{673357473152} a^{15} + \frac{46550033}{6474591088} a^{13} - \frac{153683449}{12024240592} a^{11} + \frac{148716655}{10521210518} a^{9} + \frac{682269079}{21042421036} a^{7} - \frac{402203244}{5260605259} a^{5} + \frac{418113485}{5260605259} a^{3} + \frac{2115234932}{5260605259} a$

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

\(\Q(\sqrt{2}) \), 3.1.239.1, 5.1.57121.1, 6.2.29245952.1, 10.2.106915713548288.1, 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{3}$ $30$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $30$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $30$ $15^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{15}$

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
239Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1912.2t1.b.a$1$ $ 2^{3} \cdot 239 $ \(\Q(\sqrt{-478}) \) $C_2$ (as 2T1) $1$ $-1$
1.239.2t1.a.a$1$ $ 239 $ \(\Q(\sqrt{-239}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
* 2.15296.6t3.a.a$2$ $ 2^{6} \cdot 239 $ 6.0.6989782528.1 $D_{6}$ (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.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) $1$ $0$
* 2.15296.10t3.a.a$2$ $ 2^{6} \cdot 239 $ 10.0.25552855538040832.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.15296.10t3.a.b$2$ $ 2^{6} \cdot 239 $ 10.0.25552855538040832.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.239.15t2.a.b$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.15296.30t14.a.b$2$ $ 2^{6} \cdot 239 $ 30.2.69810446891843341455650582518642209772915392512.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.239.15t2.a.d$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.15296.30t14.a.a$2$ $ 2^{6} \cdot 239 $ 30.2.69810446891843341455650582518642209772915392512.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.239.15t2.a.c$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.15296.30t14.a.d$2$ $ 2^{6} \cdot 239 $ 30.2.69810446891843341455650582518642209772915392512.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.239.15t2.a.a$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.15296.30t14.a.c$2$ $ 2^{6} \cdot 239 $ 30.2.69810446891843341455650582518642209772915392512.1 $D_{30}$ (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 *.