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Properties

Label 12.12.1556165536...0000.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{12}\cdot 3^{12}\cdot 5^{11}\cdot 11^{4}$
Root discriminant $58.35$
Ramified primes $2, 3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times C_3:S_3.C_2$ (as 12T119)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![590480, 2032800, 2504700, 1215280, 17985, -178200, -39450, 8040, 3015, -120, -90, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 90*x^10 - 120*x^9 + 3015*x^8 + 8040*x^7 - 39450*x^6 - 178200*x^5 + 17985*x^4 + 1215280*x^3 + 2504700*x^2 + 2032800*x + 590480)
 
gp: K = bnfinit(x^12 - 90*x^10 - 120*x^9 + 3015*x^8 + 8040*x^7 - 39450*x^6 - 178200*x^5 + 17985*x^4 + 1215280*x^3 + 2504700*x^2 + 2032800*x + 590480, 1)
 

Normalized defining polynomial

\( x^{12} - 90 x^{10} - 120 x^{9} + 3015 x^{8} + 8040 x^{7} - 39450 x^{6} - 178200 x^{5} + 17985 x^{4} + 1215280 x^{3} + 2504700 x^{2} + 2032800 x + 590480 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1556165536200000000000=2^{12}\cdot 3^{12}\cdot 5^{11}\cdot 11^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.35$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{44} a^{9} - \frac{1}{22} a^{7} + \frac{1}{44} a^{6} - \frac{5}{22} a^{5} - \frac{1}{44} a^{4} + \frac{7}{44} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{44} a^{10} - \frac{1}{22} a^{8} + \frac{1}{44} a^{7} + \frac{1}{44} a^{6} - \frac{1}{44} a^{5} - \frac{1}{11} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{55134189352} a^{11} + \frac{1069647}{6891773669} a^{10} - \frac{230331559}{27567094676} a^{9} - \frac{141442957}{6891773669} a^{8} + \frac{989996275}{55134189352} a^{7} - \frac{958916797}{13783547338} a^{6} - \frac{5571422255}{27567094676} a^{5} - \frac{2921514649}{13783547338} a^{4} + \frac{15882515345}{55134189352} a^{3} + \frac{475413909}{1253049758} a^{2} + \frac{278954547}{1253049758} a - \frac{196706901}{626524879}$

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:  $11$
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:  \( 7041584.90614 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times C_3:S_3.C_2$ (as 12T119):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 216
The 18 conjugacy class representatives for $S_3\times C_3:S_3.C_2$
Character table for $S_3\times C_3:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\)

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 siblings: 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.12.12.19$x^{12} + 33 x^{11} - 9 x^{10} - 18 x^{9} - 18 x^{8} + 81 x^{7} - 63 x^{6} + 108 x^{5} - 54 x^{4} + 81 x^{3} + 81 x^{2} + 81 x - 81$$3$$4$$12$12T119$[3/2, 3/2, 3/2]_{2}^{4}$
$5$5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
$11$11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$