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Properties

Label 18.0.71258351338...0103.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{36}\cdot 7^{15}$
Root discriminant $45.55$
Ramified primes $3, 7$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_2\times C_3:S_3$ (as 18T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1331, 0, 0, 228, 0, 0, 891, 0, 0, 680, 0, 0, 81, 0, 0, -12, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^15 + 81*x^12 + 680*x^9 + 891*x^6 + 228*x^3 + 1331)
 
gp: K = bnfinit(x^18 - 12*x^15 + 81*x^12 + 680*x^9 + 891*x^6 + 228*x^3 + 1331, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{15} + 81 x^{12} + 680 x^{9} + 891 x^{6} + 228 x^{3} + 1331 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-712583513384965051151503760103=-\,3^{36}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.55$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{6} - \frac{1}{8} a^{3} + \frac{1}{8}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{7} - \frac{1}{8} a^{4} + \frac{1}{8} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{5} + \frac{1}{8} a^{2}$, $\frac{1}{32} a^{12} + \frac{1}{16} a^{6} - \frac{3}{32}$, $\frac{1}{32} a^{13} + \frac{1}{16} a^{7} - \frac{3}{32} a$, $\frac{1}{32} a^{14} + \frac{1}{16} a^{8} - \frac{3}{32} a^{2}$, $\frac{1}{380480} a^{15} - \frac{5791}{380480} a^{12} + \frac{781}{38048} a^{9} - \frac{4643}{38048} a^{6} - \frac{2111}{13120} a^{3} + \frac{128669}{380480}$, $\frac{1}{4185280} a^{16} - \frac{53351}{4185280} a^{13} - \frac{3975}{418528} a^{10} + \frac{38161}{418528} a^{7} + \frac{2939}{13120} a^{4} + \frac{318909}{4185280} a$, $\frac{1}{46038080} a^{17} + \frac{208229}{46038080} a^{14} + \frac{257605}{4603808} a^{11} + \frac{352057}{4603808} a^{8} + \frac{34099}{144320} a^{5} - \frac{7266911}{46038080} a^{2}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  $8$
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:  \( 102332590.90594403 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3$ (as 18T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_2\times C_3:S_3$
Character table for $C_2\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.11907.2, 3.1.243.1, 3.1.1323.1, 3.1.11907.1, 6.0.992436543.3, 6.0.12252303.1, 6.0.20253807.1, 6.0.992436543.1, 9.1.45579633110361.5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 18 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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