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Properties

Label 36.0.77455827645...5632.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{107}$
Root discriminant $52.38$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times C_{18}):C_6$ (as 36T185)

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

\( x^{36} + 3 \)

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:  \(77455827645541172243429237514462094454119707909588182611525632=2^{36}\cdot 3^{107}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.38$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1}{2} a^{18} + \frac{1}{2} \) (order $6$)
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:  \( 146190247413682400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_{18}):C_6$ (as 36T185):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 216
The 31 conjugacy class representatives for $(C_2\times C_{18}):C_6$
Character table for $(C_2\times C_{18}):C_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 4.0.432.1, 6.0.177147.2, 9.1.2541865828329.2 x3, 12.0.385610460475392.3, 18.0.19383245667680019896796723.2

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 R ${\href{/LocalNumberField/5.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ $18{,}\,{\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ $18^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{9}$ $18{,}\,{\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ $18{,}\,{\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ $18{,}\,{\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$2$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.12.12.18$x^{12} + 80 x^{10} + 81 x^{8} - 160 x^{6} - 117 x^{4} + 80 x^{2} + 227$$2$$6$$12$$D_4 \times C_3$$[2, 2]^{6}$
2.12.12.18$x^{12} + 80 x^{10} + 81 x^{8} - 160 x^{6} - 117 x^{4} + 80 x^{2} + 227$$2$$6$$12$$D_4 \times C_3$$[2, 2]^{6}$
3Data not computed