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Properties

Label 12.0.1586874322944.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.587\times 10^{12}$
Root discriminant \(10.39\)
Ramified primes see page
Class number $1$
Class group trivial
Galois group $C_6\times C_2$ (as 12T2)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^6 + 1)
 
gp: K = bnfinit(x^12 - x^6 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1]);
 

\( x^{12} - x^{6} + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(1586874322944\) \(\medspace = 2^{12}\cdot 3^{18}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(10.39\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:   \(2\), \(3\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Gal(K/\Q) }$:  $12$
This field is Galois and abelian over $\Q$.
Conductor:  \(36=2^{2}\cdot 3^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{36}(1,·)$, $\chi_{36}(35,·)$, $\chi_{36}(5,·)$, $\chi_{36}(7,·)$, $\chi_{36}(11,·)$, $\chi_{36}(13,·)$, $\chi_{36}(17,·)$, $\chi_{36}(19,·)$, $\chi_{36}(23,·)$, $\chi_{36}(25,·)$, $\chi_{36}(29,·)$, $\chi_{36}(31,·)$$\rbrace$
This is a CM field.
Reflex fields:  \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})\)$^{3}$, 6.0.419904.1$^{3}$, \(\Q(\zeta_{36})\)$^{24}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$ Copy content Toggle raw display

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:   \( a \)  (order $36$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $a^{4}+1$, $a^{4}-a^{2}$, $a^{3}+1$, $a-1$, $a^{11}-a^{10}+a^{4}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 162.837701397 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 162.837701397 \cdot 1}{36\sqrt{1586874322944}}\approx 0.220932906611$

Galois group

$C_2\times C_6$ (as 12T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{12})\), 6.0.419904.1, \(\Q(\zeta_{9})\), \(\Q(\zeta_{36})^+\)

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 ${\href{/padicField/5.6.0.1}{6} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{6}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.1.0.1}{1} }^{12}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
\(3\) Copy content Toggle raw display 3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$