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Properties

Label 30.30.3124501777...5712.1
Degree $30$
Signature $[30, 0]$
Discriminant $2^{30}\cdot 3^{45}\cdot 11^{24}$
Root discriminant $70.77$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{30}$ (as 30T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-243, 0, 10935, 0, -151875, 0, 979290, 0, -3473442, 0, 7410609, 0, -10076778, 0, 9065763, 0, -5517450, 0, 2294766, 0, -651888, 0, 124974, 0, -15732, 0, 1233, 0, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 54*x^28 + 1233*x^26 - 15732*x^24 + 124974*x^22 - 651888*x^20 + 2294766*x^18 - 5517450*x^16 + 9065763*x^14 - 10076778*x^12 + 7410609*x^10 - 3473442*x^8 + 979290*x^6 - 151875*x^4 + 10935*x^2 - 243)
 
gp: K = bnfinit(x^30 - 54*x^28 + 1233*x^26 - 15732*x^24 + 124974*x^22 - 651888*x^20 + 2294766*x^18 - 5517450*x^16 + 9065763*x^14 - 10076778*x^12 + 7410609*x^10 - 3473442*x^8 + 979290*x^6 - 151875*x^4 + 10935*x^2 - 243, 1)
 

Normalized defining polynomial

\( x^{30} - 54 x^{28} + 1233 x^{26} - 15732 x^{24} + 124974 x^{22} - 651888 x^{20} + 2294766 x^{18} - 5517450 x^{16} + 9065763 x^{14} - 10076778 x^{12} + 7410609 x^{10} - 3473442 x^{8} + 979290 x^{6} - 151875 x^{4} + 10935 x^{2} - 243 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $30$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[30, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31245017777306374823059337284350587021473200026746355712=2^{30}\cdot 3^{45}\cdot 11^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.77$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(396=2^{2}\cdot 3^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{396}(383,·)$, $\chi_{396}(1,·)$, $\chi_{396}(323,·)$, $\chi_{396}(133,·)$, $\chi_{396}(71,·)$, $\chi_{396}(265,·)$, $\chi_{396}(203,·)$, $\chi_{396}(335,·)$, $\chi_{396}(251,·)$, $\chi_{396}(23,·)$, $\chi_{396}(25,·)$, $\chi_{396}(155,·)$, $\chi_{396}(361,·)$, $\chi_{396}(157,·)$, $\chi_{396}(287,·)$, $\chi_{396}(289,·)$, $\chi_{396}(229,·)$, $\chi_{396}(37,·)$, $\chi_{396}(97,·)$, $\chi_{396}(119,·)$, $\chi_{396}(169,·)$, $\chi_{396}(301,·)$, $\chi_{396}(47,·)$, $\chi_{396}(49,·)$, $\chi_{396}(179,·)$, $\chi_{396}(181,·)$, $\chi_{396}(311,·)$, $\chi_{396}(313,·)$, $\chi_{396}(59,·)$, $\chi_{396}(191,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{9} a^{12}$, $\frac{1}{9} a^{13}$, $\frac{1}{9} a^{14}$, $\frac{1}{9} a^{15}$, $\frac{1}{9} a^{16}$, $\frac{1}{9} a^{17}$, $\frac{1}{27} a^{18}$, $\frac{1}{27} a^{19}$, $\frac{1}{27} a^{20}$, $\frac{1}{27} a^{21}$, $\frac{1}{27} a^{22}$, $\frac{1}{27} a^{23}$, $\frac{1}{81} a^{24}$, $\frac{1}{81} a^{25}$, $\frac{1}{16119} a^{26} - \frac{94}{16119} a^{24} - \frac{61}{5373} a^{22} - \frac{88}{5373} a^{20} - \frac{8}{597} a^{18} + \frac{94}{1791} a^{16} - \frac{25}{597} a^{14} + \frac{95}{1791} a^{12} - \frac{11}{199} a^{10} + \frac{40}{597} a^{8} - \frac{95}{597} a^{6} - \frac{51}{199} a^{4} - \frac{2}{199} a^{2} + \frac{98}{199}$, $\frac{1}{16119} a^{27} - \frac{94}{16119} a^{25} - \frac{61}{5373} a^{23} - \frac{88}{5373} a^{21} - \frac{8}{597} a^{19} + \frac{94}{1791} a^{17} - \frac{25}{597} a^{15} + \frac{95}{1791} a^{13} - \frac{11}{199} a^{11} + \frac{40}{597} a^{9} - \frac{95}{597} a^{7} - \frac{51}{199} a^{5} - \frac{2}{199} a^{3} + \frac{98}{199} a$, $\frac{1}{21903478389465471} a^{28} + \frac{154081532849}{7301159463155157} a^{26} - \frac{131010207677525}{21903478389465471} a^{24} - \frac{85349501463760}{7301159463155157} a^{22} - \frac{59640972579644}{7301159463155157} a^{20} + \frac{1786084798342}{7301159463155157} a^{18} - \frac{128066173696048}{2433719821051719} a^{16} + \frac{102265829605355}{2433719821051719} a^{14} - \frac{24624299898893}{2433719821051719} a^{12} + \frac{111557765609594}{811239940350573} a^{10} - \frac{64682639803519}{811239940350573} a^{8} - \frac{4174442717978}{811239940350573} a^{6} - \frac{107202787826711}{270413313450191} a^{4} + \frac{48845719161674}{270413313450191} a^{2} + \frac{23632150421837}{270413313450191}$, $\frac{1}{21903478389465471} a^{29} + \frac{154081532849}{7301159463155157} a^{27} - \frac{131010207677525}{21903478389465471} a^{25} - \frac{85349501463760}{7301159463155157} a^{23} - \frac{59640972579644}{7301159463155157} a^{21} + \frac{1786084798342}{7301159463155157} a^{19} - \frac{128066173696048}{2433719821051719} a^{17} + \frac{102265829605355}{2433719821051719} a^{15} - \frac{24624299898893}{2433719821051719} a^{13} + \frac{111557765609594}{811239940350573} a^{11} - \frac{64682639803519}{811239940350573} a^{9} - \frac{4174442717978}{811239940350573} a^{7} - \frac{107202787826711}{270413313450191} a^{5} + \frac{48845719161674}{270413313450191} a^{3} + \frac{23632150421837}{270413313450191} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $29$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1288437153447097900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{30}$ (as 30T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 30
The 30 conjugacy class representatives for $C_{30}$
Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{36})^+\), 10.10.53339349076992.1, 15.15.10943023107606534329121.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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 $30$ $30$ R $15^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$ $30$ $30$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{6}$ $30$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$ $15^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{3}$ $15^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
3Data not computed
$11$11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$
11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$