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Properties

Label 30.0.290...763.1
Degree $30$
Signature $[0, 15]$
Discriminant $-2.910\times 10^{46}$
Root discriminant $35.38$
Ramified primes $3, 11$
Class number $31$ (GRH)
Class group $[31]$ (GRH)
Galois group $C_{30}$ (as 30T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 4*x^27 + 47*x^24 + 4*x^21 + 1159*x^18 - 1525*x^15 + 4898*x^12 + 2458*x^9 + 1824*x^6 - 42*x^3 + 1)
 
gp: K = bnfinit(x^30 - 4*x^27 + 47*x^24 + 4*x^21 + 1159*x^18 - 1525*x^15 + 4898*x^12 + 2458*x^9 + 1824*x^6 - 42*x^3 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -42, 0, 0, 1824, 0, 0, 2458, 0, 0, 4898, 0, 0, -1525, 0, 0, 1159, 0, 0, 4, 0, 0, 47, 0, 0, -4, 0, 0, 1]);
 

\(x^{30} - 4 x^{27} + 47 x^{24} + 4 x^{21} + 1159 x^{18} - 1525 x^{15} + 4898 x^{12} + 2458 x^{9} + 1824 x^{6} - 42 x^{3} + 1\)  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:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-29099190400267368949073680941341991556317731763\)\(\medspace = -\,3^{45}\cdot 11^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $35.38$
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:  $3, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $30$
This field is Galois and abelian over $\Q$.
Conductor:  \(99=3^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{99}(64,·)$, $\chi_{99}(1,·)$, $\chi_{99}(67,·)$, $\chi_{99}(4,·)$, $\chi_{99}(5,·)$, $\chi_{99}(70,·)$, $\chi_{99}(71,·)$, $\chi_{99}(14,·)$, $\chi_{99}(16,·)$, $\chi_{99}(82,·)$, $\chi_{99}(20,·)$, $\chi_{99}(86,·)$, $\chi_{99}(23,·)$, $\chi_{99}(25,·)$, $\chi_{99}(26,·)$, $\chi_{99}(91,·)$, $\chi_{99}(92,·)$, $\chi_{99}(31,·)$, $\chi_{99}(80,·)$, $\chi_{99}(34,·)$, $\chi_{99}(37,·)$, $\chi_{99}(38,·)$, $\chi_{99}(97,·)$, $\chi_{99}(47,·)$, $\chi_{99}(49,·)$, $\chi_{99}(53,·)$, $\chi_{99}(89,·)$, $\chi_{99}(56,·)$, $\chi_{99}(58,·)$, $\chi_{99}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{13333} a^{24} + \frac{1046}{13333} a^{21} - \frac{4153}{13333} a^{18} - \frac{4586}{13333} a^{15} - \frac{3958}{13333} a^{12} - \frac{6124}{13333} a^{9} + \frac{5299}{13333} a^{6} + \frac{5344}{13333} a^{3} - \frac{393}{13333}$, $\frac{1}{13333} a^{25} + \frac{1046}{13333} a^{22} - \frac{4153}{13333} a^{19} - \frac{4586}{13333} a^{16} - \frac{3958}{13333} a^{13} - \frac{6124}{13333} a^{10} + \frac{5299}{13333} a^{7} + \frac{5344}{13333} a^{4} - \frac{393}{13333} a$, $\frac{1}{13333} a^{26} + \frac{1046}{13333} a^{23} - \frac{4153}{13333} a^{20} - \frac{4586}{13333} a^{17} - \frac{3958}{13333} a^{14} - \frac{6124}{13333} a^{11} + \frac{5299}{13333} a^{8} + \frac{5344}{13333} a^{5} - \frac{393}{13333} a^{2}$, $\frac{1}{5562777263664919} a^{27} - \frac{157248114464}{5562777263664919} a^{24} + \frac{195426713654710}{5562777263664919} a^{21} - \frac{312298975194891}{5562777263664919} a^{18} - \frac{2160781218433615}{5562777263664919} a^{15} + \frac{1365544977074574}{5562777263664919} a^{12} + \frac{918911329346005}{5562777263664919} a^{9} + \frac{483514846846244}{5562777263664919} a^{6} - \frac{1574124489529863}{5562777263664919} a^{3} - \frac{1801075505016551}{5562777263664919}$, $\frac{1}{5562777263664919} a^{28} - \frac{157248114464}{5562777263664919} a^{25} + \frac{195426713654710}{5562777263664919} a^{22} - \frac{312298975194891}{5562777263664919} a^{19} - \frac{2160781218433615}{5562777263664919} a^{16} + \frac{1365544977074574}{5562777263664919} a^{13} + \frac{918911329346005}{5562777263664919} a^{10} + \frac{483514846846244}{5562777263664919} a^{7} - \frac{1574124489529863}{5562777263664919} a^{4} - \frac{1801075505016551}{5562777263664919} a$, $\frac{1}{5562777263664919} a^{29} - \frac{157248114464}{5562777263664919} a^{26} + \frac{195426713654710}{5562777263664919} a^{23} - \frac{312298975194891}{5562777263664919} a^{20} - \frac{2160781218433615}{5562777263664919} a^{17} + \frac{1365544977074574}{5562777263664919} a^{14} + \frac{918911329346005}{5562777263664919} a^{11} + \frac{483514846846244}{5562777263664919} a^{8} - \frac{1574124489529863}{5562777263664919} a^{5} - \frac{1801075505016551}{5562777263664919} a^{2}$  Toggle raw display

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

Class group and class number

$C_{31}$, which has order $31$ (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:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{430403920418428}{5562777263664919} a^{28} + \frac{1710665660120153}{5562777263664919} a^{25} - \frac{20186503940073518}{5562777263664919} a^{22} - \frac{2229897514199135}{5562777263664919} a^{19} - \frac{498949407475743763}{5562777263664919} a^{16} + \frac{643722071295932200}{5562777263664919} a^{13} - \frac{2092985376211195994}{5562777263664919} a^{10} - \frac{1108399926852478656}{5562777263664919} a^{7} - \frac{821994629375481848}{5562777263664919} a^{4} - \frac{615014390609465}{5562777263664919} a \) (order $18$)  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:  \( 15853905121.091976 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 15853905121.091976 \cdot 31}{18\sqrt{29099190400267368949073680941341991556317731763}}\approx 0.150308318304457$ (assuming GRH)

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{9})\), 10.0.52089208083.1, 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 $30$ R $30$ $15^{2}$ R $15^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ $30$ $15^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{6}$ $30$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{10}$ $30$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{3}$ $30$

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
3Data not computed
11Data not computed