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Properties

Label 36.0.13106567101...9081.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{54}\cdot 7^{30}$
Root discriminant $26.30$
Ramified primes $3, 7$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1)
 
gp: K = bnfinit(x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{33} + x^{27} - x^{24} + x^{18} - x^{12} + x^{9} - x^{3} + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1310656710125779295611091389185381649163216325999081=3^{54}\cdot 7^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.30$
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:  $3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(63=3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{63}(1,·)$, $\chi_{63}(2,·)$, $\chi_{63}(4,·)$, $\chi_{63}(5,·)$, $\chi_{63}(8,·)$, $\chi_{63}(10,·)$, $\chi_{63}(11,·)$, $\chi_{63}(13,·)$, $\chi_{63}(16,·)$, $\chi_{63}(17,·)$, $\chi_{63}(19,·)$, $\chi_{63}(20,·)$, $\chi_{63}(22,·)$, $\chi_{63}(23,·)$, $\chi_{63}(25,·)$, $\chi_{63}(26,·)$, $\chi_{63}(29,·)$, $\chi_{63}(31,·)$, $\chi_{63}(32,·)$, $\chi_{63}(34,·)$, $\chi_{63}(37,·)$, $\chi_{63}(38,·)$, $\chi_{63}(40,·)$, $\chi_{63}(41,·)$, $\chi_{63}(43,·)$, $\chi_{63}(44,·)$, $\chi_{63}(46,·)$, $\chi_{63}(47,·)$, $\chi_{63}(50,·)$, $\chi_{63}(52,·)$, $\chi_{63}(53,·)$, $\chi_{63}(55,·)$, $\chi_{63}(58,·)$, $\chi_{63}(59,·)$, $\chi_{63}(61,·)$, $\chi_{63}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -a \) (order $126$)
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:  \( 408184875586.38837 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2$ (as 36T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{9})^+\), 3.3.3969.2, 3.3.3969.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\zeta_{7})\), 6.0.2250423.1, 6.0.110270727.1, 6.0.110270727.2, 6.0.64827.1, \(\Q(\zeta_{21})^+\), \(\Q(\zeta_{9})\), 6.6.6751269.1, 6.0.47258883.1, 6.6.330812181.1, 6.0.47258883.2, 6.6.330812181.2, 9.9.62523502209.1, \(\Q(\zeta_{21})\), 12.0.45579633110361.1, 12.0.109436699097976761.2, 12.0.109436699097976761.1, 18.0.1340851596668237962730583.1, 18.0.105548084868928352751387.1, \(\Q(\zeta_{63})^+\)

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
3Data not computed
7Data not computed