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Properties

Label 12.2.77998046721343488.26
Degree $12$
Signature $[2, 5]$
Discriminant $-\,2^{26}\cdot 3^{19}$
Root discriminant $25.57$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $D_6:D_6$ (as 12T81)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6, 0, 36, -12, 18, 0, -6, -36, -15, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 15*x^8 - 36*x^7 - 6*x^6 + 18*x^4 - 12*x^3 + 36*x^2 + 6)
 
gp: K = bnfinit(x^12 - 15*x^8 - 36*x^7 - 6*x^6 + 18*x^4 - 12*x^3 + 36*x^2 + 6, 1)
 

Normalized defining polynomial

\( x^{12} - 15 x^{8} - 36 x^{7} - 6 x^{6} + 18 x^{4} - 12 x^{3} + 36 x^{2} + 6 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-77998046721343488=-\,2^{26}\cdot 3^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.57$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{1738} a^{10} - \frac{51}{869} a^{9} - \frac{765}{1738} a^{8} + \frac{334}{869} a^{7} - \frac{46}{869} a^{6} - \frac{369}{869} a^{5} - \frac{406}{869} a^{4} + \frac{263}{869} a^{3} + \frac{294}{869} a^{2} + \frac{194}{869} a - \frac{387}{869}$, $\frac{1}{497068} a^{11} - \frac{63}{497068} a^{10} - \frac{220255}{497068} a^{9} - \frac{36119}{497068} a^{8} - \frac{7589}{22594} a^{7} - \frac{42137}{248534} a^{6} + \frac{22582}{124267} a^{5} + \frac{56955}{124267} a^{4} + \frac{102665}{248534} a^{3} - \frac{6761}{22594} a^{2} - \frac{97101}{248534} a - \frac{101993}{248534}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $6$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 28739.8558417 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_6:D_6$ (as 12T81):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 24 conjugacy class representatives for $D_6:D_6$
Character table for $D_6:D_6$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.2.6912.1, 6.2.20155392.2

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

Sibling fields

Degree 12 siblings: data not computed
Degree 24 siblings: 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 ${\href{/LocalNumberField/5.12.0.1}{12} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.4.8.5$x^{4} + 2 x^{2} + 4 x + 6$$4$$1$$8$$D_{4}$$[2, 3]^{2}$
2.8.18.54$x^{8} + 6 x^{6} + 4 x^{3} + 6$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
$3$3.12.19.26$x^{12} + 6 x^{11} - 9 x^{10} + 12 x^{9} + 12 x^{8} - 9 x^{7} + 3 x^{6} - 9 x^{5} + 9 x^{4} + 6 x^{3} - 9 x^{2} - 9 x + 6$$12$$1$$19$12T42$[3/2, 2]_{4}^{2}$