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Properties

Label 30.30.1123215710...4704.1
Degree $30$
Signature $[30, 0]$
Discriminant $2^{30}\cdot 7^{20}\cdot 11^{27}$
Root discriminant $63.34$
Ramified primes $2, 7, 11$
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![-1331, 0, 39930, 0, -408617, 0, 2110966, 0, -6411427, 0, 12328085, 0, -15598836, 0, 13252283, 0, -7637278, 0, 3000195, 0, -803154, 0, 145222, 0, -17325, 0, 1298, 0, -55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 55*x^28 + 1298*x^26 - 17325*x^24 + 145222*x^22 - 803154*x^20 + 3000195*x^18 - 7637278*x^16 + 13252283*x^14 - 15598836*x^12 + 12328085*x^10 - 6411427*x^8 + 2110966*x^6 - 408617*x^4 + 39930*x^2 - 1331)
 
gp: K = bnfinit(x^30 - 55*x^28 + 1298*x^26 - 17325*x^24 + 145222*x^22 - 803154*x^20 + 3000195*x^18 - 7637278*x^16 + 13252283*x^14 - 15598836*x^12 + 12328085*x^10 - 6411427*x^8 + 2110966*x^6 - 408617*x^4 + 39930*x^2 - 1331, 1)
 

Normalized defining polynomial

\( x^{30} - 55 x^{28} + 1298 x^{26} - 17325 x^{24} + 145222 x^{22} - 803154 x^{20} + 3000195 x^{18} - 7637278 x^{16} + 13252283 x^{14} - 15598836 x^{12} + 12328085 x^{10} - 6411427 x^{8} + 2110966 x^{6} - 408617 x^{4} + 39930 x^{2} - 1331 \)

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:  \(1123215710861738275456915198522447563874269573052104704=2^{30}\cdot 7^{20}\cdot 11^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.34$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(308=2^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{308}(1,·)$, $\chi_{308}(107,·)$, $\chi_{308}(263,·)$, $\chi_{308}(9,·)$, $\chi_{308}(183,·)$, $\chi_{308}(239,·)$, $\chi_{308}(141,·)$, $\chi_{308}(79,·)$, $\chi_{308}(81,·)$, $\chi_{308}(211,·)$, $\chi_{308}(151,·)$, $\chi_{308}(225,·)$, $\chi_{308}(25,·)$, $\chi_{308}(219,·)$, $\chi_{308}(93,·)$, $\chi_{308}(95,·)$, $\chi_{308}(289,·)$, $\chi_{308}(113,·)$, $\chi_{308}(37,·)$, $\chi_{308}(39,·)$, $\chi_{308}(169,·)$, $\chi_{308}(43,·)$, $\chi_{308}(303,·)$, $\chi_{308}(177,·)$, $\chi_{308}(221,·)$, $\chi_{308}(51,·)$, $\chi_{308}(53,·)$, $\chi_{308}(137,·)$, $\chi_{308}(123,·)$, $\chi_{308}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10}$, $\frac{1}{11} a^{11}$, $\frac{1}{11} a^{12}$, $\frac{1}{11} a^{13}$, $\frac{1}{11} a^{14}$, $\frac{1}{11} a^{15}$, $\frac{1}{11} a^{16}$, $\frac{1}{11} a^{17}$, $\frac{1}{11} a^{18}$, $\frac{1}{11} a^{19}$, $\frac{1}{121} a^{20}$, $\frac{1}{121} a^{21}$, $\frac{1}{121} a^{22}$, $\frac{1}{121} a^{23}$, $\frac{1}{5203} a^{24} + \frac{1}{473} a^{22} + \frac{9}{5203} a^{20} + \frac{1}{43} a^{18} + \frac{13}{473} a^{16} - \frac{5}{473} a^{14} + \frac{6}{473} a^{12} - \frac{5}{473} a^{10} + \frac{11}{43} a^{8} - \frac{1}{43} a^{6} - \frac{1}{43} a^{4} + \frac{7}{43} a^{2} - \frac{2}{43}$, $\frac{1}{5203} a^{25} + \frac{1}{473} a^{23} + \frac{9}{5203} a^{21} + \frac{1}{43} a^{19} + \frac{13}{473} a^{17} - \frac{5}{473} a^{15} + \frac{6}{473} a^{13} - \frac{5}{473} a^{11} + \frac{11}{43} a^{9} - \frac{1}{43} a^{7} - \frac{1}{43} a^{5} + \frac{7}{43} a^{3} - \frac{2}{43} a$, $\frac{1}{5203} a^{26} + \frac{17}{5203} a^{22} - \frac{21}{5203} a^{20} + \frac{21}{473} a^{18} - \frac{19}{473} a^{16} + \frac{18}{473} a^{14} + \frac{15}{473} a^{12} + \frac{4}{473} a^{10} + \frac{7}{43} a^{8} + \frac{10}{43} a^{6} + \frac{18}{43} a^{4} + \frac{7}{43} a^{2} - \frac{21}{43}$, $\frac{1}{5203} a^{27} + \frac{17}{5203} a^{23} - \frac{21}{5203} a^{21} + \frac{21}{473} a^{19} - \frac{19}{473} a^{17} + \frac{18}{473} a^{15} + \frac{15}{473} a^{13} + \frac{4}{473} a^{11} + \frac{7}{43} a^{9} + \frac{10}{43} a^{7} + \frac{18}{43} a^{5} + \frac{7}{43} a^{3} - \frac{21}{43} a$, $\frac{1}{1779433424894323} a^{28} + \frac{66936678742}{1779433424894323} a^{26} + \frac{41218577492}{1779433424894323} a^{24} + \frac{519987916271}{1779433424894323} a^{22} + \frac{1800644084939}{1779433424894323} a^{20} - \frac{323809725138}{161766674990393} a^{18} + \frac{5943010899553}{161766674990393} a^{16} + \frac{5851142472044}{161766674990393} a^{14} + \frac{1055330608364}{161766674990393} a^{12} + \frac{3864204640248}{161766674990393} a^{10} + \frac{2496246443913}{14706061362763} a^{8} + \frac{211319476237}{14706061362763} a^{6} - \frac{7004197987566}{14706061362763} a^{4} - \frac{4292222891592}{14706061362763} a^{2} + \frac{729743559581}{14706061362763}$, $\frac{1}{1779433424894323} a^{29} + \frac{66936678742}{1779433424894323} a^{27} + \frac{41218577492}{1779433424894323} a^{25} + \frac{519987916271}{1779433424894323} a^{23} + \frac{1800644084939}{1779433424894323} a^{21} - \frac{323809725138}{161766674990393} a^{19} + \frac{5943010899553}{161766674990393} a^{17} + \frac{5851142472044}{161766674990393} a^{15} + \frac{1055330608364}{161766674990393} a^{13} + \frac{3864204640248}{161766674990393} a^{11} + \frac{2496246443913}{14706061362763} a^{9} + \frac{211319476237}{14706061362763} a^{7} - \frac{7004197987566}{14706061362763} a^{5} - \frac{4292222891592}{14706061362763} a^{3} + \frac{729743559581}{14706061362763} a$

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:  \( 399416357360943940 \) (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

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.6.204526784.1, \(\Q(\zeta_{44})^+\), 15.15.886528337182930278529.1

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 $30$ $15^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{3}$ $30$ $15^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ $30$ $15^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{30}$ $30$ $15^{2}$ $30$

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
$7$7.15.10.1$x^{15} + 4116 x^{6} - 2401 x^{3} + 1075648$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
7.15.10.1$x^{15} + 4116 x^{6} - 2401 x^{3} + 1075648$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
11Data not computed