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Properties

Label 30.0.62494086770...1472.1
Degree $30$
Signature $[0, 15]$
Discriminant $-\,2^{45}\cdot 31^{29}$
Root discriminant $78.20$
Ramified primes $2, 31$
Class number $2302432$ (GRH)
Class group $[2, 2, 575608]$ (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![1015808, 0, 20316160, 0, 120881152, 0, 336740352, 0, 533172224, 0, 533172224, 0, 358865920, 0, 169179648, 0, 57222528, 0, 14054656, 0, 2509760, 0, 322400, 0, 29016, 0, 1736, 0, 62, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 62*x^28 + 1736*x^26 + 29016*x^24 + 322400*x^22 + 2509760*x^20 + 14054656*x^18 + 57222528*x^16 + 169179648*x^14 + 358865920*x^12 + 533172224*x^10 + 533172224*x^8 + 336740352*x^6 + 120881152*x^4 + 20316160*x^2 + 1015808)
 
gp: K = bnfinit(x^30 + 62*x^28 + 1736*x^26 + 29016*x^24 + 322400*x^22 + 2509760*x^20 + 14054656*x^18 + 57222528*x^16 + 169179648*x^14 + 358865920*x^12 + 533172224*x^10 + 533172224*x^8 + 336740352*x^6 + 120881152*x^4 + 20316160*x^2 + 1015808, 1)
 

Normalized defining polynomial

\( x^{30} + 62 x^{28} + 1736 x^{26} + 29016 x^{24} + 322400 x^{22} + 2509760 x^{20} + 14054656 x^{18} + 57222528 x^{16} + 169179648 x^{14} + 358865920 x^{12} + 533172224 x^{10} + 533172224 x^{8} + 336740352 x^{6} + 120881152 x^{4} + 20316160 x^{2} + 1015808 \)

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:  \(-624940867704923870335005165628439523412241657929737961472=-\,2^{45}\cdot 31^{29}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.20$
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$$\chi_{248}(1,·)$, $\chi_{248}(213,·)$, $\chi_{248}(197,·)$, $\chi_{248}(129,·)$, $\chi_{248}(9,·)$, $\chi_{248}(13,·)$, $\chi_{248}(141,·)$, $\chi_{248}(77,·)$, $\chi_{248}(81,·)$, $\chi_{248}(85,·)$, $\chi_{248}(21,·)$, $\chi_{248}(25,·)$, $\chi_{248}(29,·)$, $\chi_{248}(229,·)$, $\chi_{248}(33,·)$, $\chi_{248}(113,·)$, $\chi_{248}(37,·)$, $\chi_{248}(97,·)$, $\chi_{248}(41,·)$, $\chi_{248}(193,·)$, $\chi_{248}(225,·)$, $\chi_{248}(189,·)$, $\chi_{248}(49,·)$, $\chi_{248}(181,·)$, $\chi_{248}(233,·)$, $\chi_{248}(121,·)$, $\chi_{248}(117,·)$, $\chi_{248}(61,·)$, $\chi_{248}(169,·)$, $\chi_{248}(53,·)$$\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

$C_{2}\times C_{2}\times C_{575608}$, which has order $2302432$ (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

\(\Q(\sqrt{-62}) \), 3.3.961.1, 5.5.923521.1, 6.0.14658125312.1, 10.0.866373538960867328.1, \(\Q(\zeta_{31})^+\)

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 $15^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{5}$ $15^{2}$ $15^{2}$ $15^{2}$ $30$ $30$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{10}$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{6}$ $15^{2}$ $30$

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
$2$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
31Data not computed