Properties

Label 12.0.6053445140625.2
Degree $12$
Signature $[0, 6]$
Discriminant $6.053\times 10^{12}$
Root discriminant \(11.62\)
Ramified primes see page
Class number $1$
Class group trivial
Galois group $C_6\times S_3$ (as 12T18)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^9 + 2*x^6 + x^3 + 1)
 
gp: K = bnfinit(x^12 - x^9 + 2*x^6 + x^3 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 1, 0, 0, 2, 0, 0, -1, 0, 0, 1]);
 

\( x^{12} - x^{9} + 2x^{6} + x^{3} + 1 \) 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:   \(6053445140625\) \(\medspace = 3^{18}\cdot 5^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(11.62\)
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:   \(3\), \(5\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $6$
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

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:   \( \frac{1}{2} a^{9} - a^{6} + a^{3} + \frac{1}{2} \)  (order $6$) 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:   $\frac{1}{2}a^{10}-a^{7}+a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+a^{7}-a^{4}+\frac{3}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $a^{11}-\frac{1}{2}a^{10}-a^{9}-a^{8}+a^{7}+a^{6}+2a^{5}-a^{4}-2a^{3}+a^{2}+\frac{1}{2}a-1$, $\frac{1}{2}a^{11}+a^{10}+\frac{1}{2}a^{9}-a^{8}-a^{7}+2a^{5}+2a^{4}-\frac{3}{2}a^{2}+a+\frac{3}{2}$, $\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-a^{7}+a^{4}-a^{3}+\frac{3}{2}a^{2}-\frac{1}{2}a$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 106.294858375 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 106.294858375 \cdot 1}{6\sqrt{6053445140625}}\approx 0.443035892439$

Galois group

$C_6\times S_3$ (as 12T18):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 36
The 18 conjugacy class representatives for $C_6\times S_3$
Character table for $C_6\times S_3$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.2460375.2

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

Sibling fields

Galois closure: Deg 36
Degree 18 siblings: 18.6.402131117372361328125.1, 18.0.9651146816936671875.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }^{3}$ R R ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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
\(3\) Copy content Toggle raw display 3.12.18.35$x^{12} - 39 x^{11} - 12 x^{9} + 27 x^{8} - 27 x^{7} + 18 x^{6} - 9 x^{5} - 36 x^{4} - 27 x^{3} - 27 x^{2} + 27 x + 36$$6$$2$$18$$C_6\times S_3$$[3/2, 2]_{2}^{2}$
\(5\) Copy content Toggle raw display 5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.15.2t1.a.a$1$ $ 3 \cdot 5 $ \(\Q(\sqrt{-15}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.45.6t1.a.a$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
1.9.6t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})\) $C_6$ (as 6T1) $0$ $-1$
1.9.6t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})\) $C_6$ (as 6T1) $0$ $-1$
1.45.6t1.b.a$1$ $ 3^{2} \cdot 5 $ 6.0.2460375.1 $C_6$ (as 6T1) $0$ $-1$
1.45.6t1.a.b$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
1.45.6t1.b.b$1$ $ 3^{2} \cdot 5 $ 6.0.2460375.1 $C_6$ (as 6T1) $0$ $-1$
1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
2.135.3t2.b.a$2$ $ 3^{3} \cdot 5 $ 3.1.135.1 $S_3$ (as 3T2) $1$ $0$
2.135.6t3.b.a$2$ $ 3^{3} \cdot 5 $ 6.0.54675.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.405.6t5.a.a$2$ $ 3^{4} \cdot 5 $ 6.0.2460375.2 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.405.6t5.a.b$2$ $ 3^{4} \cdot 5 $ 6.0.2460375.2 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.405.12t18.a.a$2$ $ 3^{4} \cdot 5 $ 12.0.6053445140625.2 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.405.12t18.a.b$2$ $ 3^{4} \cdot 5 $ 12.0.6053445140625.2 $C_6\times S_3$ (as 12T18) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.