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Properties

Label 30.0.20159382829...8112.1
Degree $30$
Signature $[0, 15]$
Discriminant $-\,2^{45}\cdot 31^{28}$
Root discriminant $69.74$
Ramified primes $2, 31$
Class number $232364$ (GRH)
Class group $[2, 116182]$ (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![32768, 0, 1966080, 0, 19496960, 0, 76038144, 0, 154791936, 0, 189190144, 0, 150492160, 0, 81861120, 0, 31380096, 0, 8614144, 0, 1700160, 0, 239200, 0, 23400, 0, 1512, 0, 58, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 58*x^28 + 1512*x^26 + 23400*x^24 + 239200*x^22 + 1700160*x^20 + 8614144*x^18 + 31380096*x^16 + 81861120*x^14 + 150492160*x^12 + 189190144*x^10 + 154791936*x^8 + 76038144*x^6 + 19496960*x^4 + 1966080*x^2 + 32768)
 
gp: K = bnfinit(x^30 + 58*x^28 + 1512*x^26 + 23400*x^24 + 239200*x^22 + 1700160*x^20 + 8614144*x^18 + 31380096*x^16 + 81861120*x^14 + 150492160*x^12 + 189190144*x^10 + 154791936*x^8 + 76038144*x^6 + 19496960*x^4 + 1966080*x^2 + 32768, 1)
 

Normalized defining polynomial

\( x^{30} + 58 x^{28} + 1512 x^{26} + 23400 x^{24} + 239200 x^{22} + 1700160 x^{20} + 8614144 x^{18} + 31380096 x^{16} + 81861120 x^{14} + 150492160 x^{12} + 189190144 x^{10} + 154791936 x^{8} + 76038144 x^{6} + 19496960 x^{4} + 1966080 x^{2} + 32768 \)

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

Invariants

Degree:  $30$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 15]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-20159382829191092591451779536401274948781988965475418112=-\,2^{45}\cdot 31^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.74$
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:  $2, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(248=2^{3}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{248}(1,·)$, $\chi_{248}(131,·)$, $\chi_{248}(227,·)$, $\chi_{248}(211,·)$, $\chi_{248}(129,·)$, $\chi_{248}(9,·)$, $\chi_{248}(81,·)$, $\chi_{248}(19,·)$, $\chi_{248}(171,·)$, $\chi_{248}(225,·)$, $\chi_{248}(25,·)$, $\chi_{248}(219,·)$, $\chi_{248}(107,·)$, $\chi_{248}(33,·)$, $\chi_{248}(67,·)$, $\chi_{248}(35,·)$, $\chi_{248}(113,·)$, $\chi_{248}(163,·)$, $\chi_{248}(195,·)$, $\chi_{248}(97,·)$, $\chi_{248}(41,·)$, $\chi_{248}(193,·)$, $\chi_{248}(235,·)$, $\chi_{248}(49,·)$, $\chi_{248}(51,·)$, $\chi_{248}(187,·)$, $\chi_{248}(233,·)$, $\chi_{248}(121,·)$, $\chi_{248}(59,·)$, $\chi_{248}(169,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$

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

Class group and class number

$C_{2}\times C_{116182}$, which has order $232364$ (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:  $14$
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:  \( 4316173757.895952 \) (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

\(\Q(\sqrt{-2}) \), 3.3.961.1, 5.5.923521.1, 6.0.472842752.1, 10.0.27947533514866688.1, \(\Q(\zeta_{31})^+\)

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 $15^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{5}$ $30$ $15^{2}$ $30$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{5}$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ $30$ $15^{2}$

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
$2$2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
31Data not computed