Properties

Label 30.2.347...912.1
Degree $30$
Signature $[2, 14]$
Discriminant $3.474\times 10^{50}$
Root discriminant $48.38$
Ramified primes $2, 439$
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 - 6*x^28 + 68*x^26 - 536*x^24 + 1904*x^22 - 4192*x^20 + 2240*x^18 + 23936*x^16 - 82432*x^14 + 138752*x^12 + 235520*x^10 - 1073152*x^8 + 2486272*x^6 - 2449408*x^4 + 2015232*x^2 - 32768)
 
gp: K = bnfinit(x^30 - 6*x^28 + 68*x^26 - 536*x^24 + 1904*x^22 - 4192*x^20 + 2240*x^18 + 23936*x^16 - 82432*x^14 + 138752*x^12 + 235520*x^10 - 1073152*x^8 + 2486272*x^6 - 2449408*x^4 + 2015232*x^2 - 32768, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32768, 0, 2015232, 0, -2449408, 0, 2486272, 0, -1073152, 0, 235520, 0, 138752, 0, -82432, 0, 23936, 0, 2240, 0, -4192, 0, 1904, 0, -536, 0, 68, 0, -6, 0, 1]);
 

\(x^{30} - 6 x^{28} + 68 x^{26} - 536 x^{24} + 1904 x^{22} - 4192 x^{20} + 2240 x^{18} + 23936 x^{16} - 82432 x^{14} + 138752 x^{12} + 235520 x^{10} - 1073152 x^{8} + 2486272 x^{6} - 2449408 x^{4} + 2015232 x^{2} - 32768\)  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:  \(347418617918033867346576267434667527455579320614912\)\(\medspace = 2^{45}\cdot 439^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $48.38$
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:  $2, 439$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{48} a^{8} - \frac{1}{3}$, $\frac{1}{48} a^{9} - \frac{1}{3} a$, $\frac{1}{96} a^{10} - \frac{1}{6} a^{2}$, $\frac{1}{96} a^{11} - \frac{1}{6} a^{3}$, $\frac{1}{192} a^{12} - \frac{1}{12} a^{4}$, $\frac{1}{192} a^{13} - \frac{1}{12} a^{5}$, $\frac{1}{1152} a^{14} - \frac{1}{576} a^{12} + \frac{1}{288} a^{10} - \frac{1}{144} a^{8} - \frac{1}{72} a^{6} + \frac{1}{36} a^{4} - \frac{1}{18} a^{2} + \frac{1}{9}$, $\frac{1}{1152} a^{15} - \frac{1}{576} a^{13} + \frac{1}{288} a^{11} - \frac{1}{144} a^{9} - \frac{1}{72} a^{7} + \frac{1}{36} a^{5} - \frac{1}{18} a^{3} + \frac{1}{9} a$, $\frac{1}{2304} a^{16} + \frac{1}{144} a^{8} - \frac{2}{9}$, $\frac{1}{2304} a^{17} + \frac{1}{144} a^{9} - \frac{2}{9} a$, $\frac{1}{4608} a^{18} + \frac{1}{288} a^{10} - \frac{1}{9} a^{2}$, $\frac{1}{4608} a^{19} + \frac{1}{288} a^{11} - \frac{1}{9} a^{3}$, $\frac{1}{9216} a^{20} + \frac{1}{576} a^{12} - \frac{1}{18} a^{4}$, $\frac{1}{9216} a^{21} + \frac{1}{576} a^{13} - \frac{1}{18} a^{5}$, $\frac{1}{55296} a^{22} - \frac{1}{27648} a^{20} + \frac{1}{13824} a^{18} - \frac{1}{6912} a^{16} + \frac{1}{3456} a^{14} - \frac{1}{1728} a^{12} + \frac{1}{864} a^{10} - \frac{1}{432} a^{8} - \frac{1}{108} a^{6} + \frac{1}{54} a^{4} - \frac{1}{27} a^{2} + \frac{2}{27}$, $\frac{1}{55296} a^{23} - \frac{1}{27648} a^{21} + \frac{1}{13824} a^{19} - \frac{1}{6912} a^{17} + \frac{1}{3456} a^{15} - \frac{1}{1728} a^{13} + \frac{1}{864} a^{11} - \frac{1}{432} a^{9} - \frac{1}{108} a^{7} + \frac{1}{54} a^{5} - \frac{1}{27} a^{3} + \frac{2}{27} a$, $\frac{1}{331776} a^{24} + \frac{1}{165888} a^{22} + \frac{1}{41472} a^{20} + \frac{1}{41472} a^{18} + \frac{1}{10368} a^{16} - \frac{1}{5184} a^{14} - \frac{1}{1296} a^{12} - \frac{11}{2592} a^{10} + \frac{1}{648} a^{8} + \frac{1}{648} a^{6} + \frac{1}{162} a^{4} + \frac{5}{81} a^{2} - \frac{5}{81}$, $\frac{1}{331776} a^{25} + \frac{1}{165888} a^{23} + \frac{1}{41472} a^{21} + \frac{1}{41472} a^{19} + \frac{1}{10368} a^{17} - \frac{1}{5184} a^{15} - \frac{1}{1296} a^{13} - \frac{11}{2592} a^{11} + \frac{1}{648} a^{9} + \frac{1}{648} a^{7} + \frac{1}{162} a^{5} + \frac{5}{81} a^{3} - \frac{5}{81} a$, $\frac{1}{417374208} a^{26} - \frac{137}{208687104} a^{24} - \frac{65}{13042944} a^{22} + \frac{19}{1410048} a^{20} + \frac{329}{6521472} a^{18} - \frac{947}{13042944} a^{16} + \frac{941}{6521472} a^{14} - \frac{2683}{1630368} a^{12} - \frac{1849}{407592} a^{10} + \frac{197}{47952} a^{8} + \frac{719}{11988} a^{6} - \frac{9697}{101898} a^{4} + \frac{5024}{50949} a^{2} + \frac{5401}{16983}$, $\frac{1}{417374208} a^{27} - \frac{137}{208687104} a^{25} - \frac{65}{13042944} a^{23} + \frac{19}{1410048} a^{21} + \frac{329}{6521472} a^{19} - \frac{947}{13042944} a^{17} + \frac{941}{6521472} a^{15} - \frac{2683}{1630368} a^{13} - \frac{1849}{407592} a^{11} + \frac{197}{47952} a^{9} + \frac{719}{11988} a^{7} - \frac{9697}{101898} a^{5} + \frac{5024}{50949} a^{3} + \frac{5401}{16983} a$, $\frac{1}{323047636992} a^{28} - \frac{113}{161523818496} a^{26} + \frac{1}{65981952} a^{24} + \frac{185729}{40380954624} a^{22} + \frac{63673}{5047619328} a^{20} + \frac{364007}{10095238656} a^{18} - \frac{263437}{2523809664} a^{16} - \frac{304201}{2523809664} a^{14} - \frac{14015}{78869052} a^{12} - \frac{795607}{630952416} a^{10} + \frac{18031}{9278712} a^{8} + \frac{2867587}{78869052} a^{6} - \frac{1612129}{13144842} a^{4} - \frac{4847656}{19717263} a^{2} + \frac{4609462}{19717263}$, $\frac{1}{323047636992} a^{29} - \frac{113}{161523818496} a^{27} + \frac{1}{65981952} a^{25} + \frac{185729}{40380954624} a^{23} + \frac{63673}{5047619328} a^{21} + \frac{364007}{10095238656} a^{19} - \frac{263437}{2523809664} a^{17} - \frac{304201}{2523809664} a^{15} - \frac{14015}{78869052} a^{13} - \frac{795607}{630952416} a^{11} + \frac{18031}{9278712} a^{9} + \frac{2867587}{78869052} a^{7} - \frac{1612129}{13144842} a^{5} - \frac{4847656}{19717263} a^{3} + \frac{4609462}{19717263} a$  Toggle raw display

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 $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed  Toggle raw display
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

\(\Q(\sqrt{2}) \), 3.1.439.1, 5.1.192721.1, 6.2.98673152.2, 10.2.1217048865701888.3, 15.1.3142328914862177479.1

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{3}$ $15^{2}$ $30$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $30$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $30$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{15}$

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
439Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
1.439.2t1.a.a$1$ $ 439 $ \(\Q(\sqrt{-439}) \) $C_2$ (as 2T1) $1$ $-1$
1.3512.2t1.b.a$1$ $ 2^{3} \cdot 439 $ \(\Q(\sqrt{-878}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.28096.6t3.a.a$2$ $ 2^{6} \cdot 439 $ 6.2.98673152.2 $D_{6}$ (as 6T3) $1$ $0$
* 2.439.3t2.a.a$2$ $ 439 $ 3.1.439.1 $S_3$ (as 3T2) $1$ $0$
* 2.439.5t2.a.a$2$ $ 439 $ 5.1.192721.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.439.5t2.a.b$2$ $ 439 $ 5.1.192721.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.28096.10t3.a.b$2$ $ 2^{6} \cdot 439 $ 10.2.1217048865701888.3 $D_{10}$ (as 10T3) $1$ $0$
* 2.28096.10t3.a.a$2$ $ 2^{6} \cdot 439 $ 10.2.1217048865701888.3 $D_{10}$ (as 10T3) $1$ $0$
* 2.439.15t2.a.c$2$ $ 439 $ 15.1.3142328914862177479.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.439.15t2.a.b$2$ $ 439 $ 15.1.3142328914862177479.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.439.15t2.a.a$2$ $ 439 $ 15.1.3142328914862177479.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.439.15t2.a.d$2$ $ 439 $ 15.1.3142328914862177479.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.28096.30t14.a.a$2$ $ 2^{6} \cdot 439 $ 30.2.347418617918033867346576267434667527455579320614912.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.28096.30t14.a.c$2$ $ 2^{6} \cdot 439 $ 30.2.347418617918033867346576267434667527455579320614912.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.28096.30t14.a.d$2$ $ 2^{6} \cdot 439 $ 30.2.347418617918033867346576267434667527455579320614912.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.28096.30t14.a.b$2$ $ 2^{6} \cdot 439 $ 30.2.347418617918033867346576267434667527455579320614912.1 $D_{30}$ (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 *.