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Properties

Label 12.0.419904000000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{8}\cdot 5^{6}$
Root discriminant $9.30$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_6\times S_3$ (as 12T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 4, -8, 21, -44, 67, -74, 62, -40, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 19*x^10 - 40*x^9 + 62*x^8 - 74*x^7 + 67*x^6 - 44*x^5 + 21*x^4 - 8*x^3 + 4*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^12 - 6*x^11 + 19*x^10 - 40*x^9 + 62*x^8 - 74*x^7 + 67*x^6 - 44*x^5 + 21*x^4 - 8*x^3 + 4*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 19 x^{10} - 40 x^{9} + 62 x^{8} - 74 x^{7} + 67 x^{6} - 44 x^{5} + 21 x^{4} - 8 x^{3} + 4 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(419904000000=2^{12}\cdot 3^{8}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $9.30$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{49} a^{11} + \frac{19}{49} a^{10} + \frac{4}{49} a^{9} + \frac{11}{49} a^{8} - \frac{6}{49} a^{7} + \frac{3}{7} a^{6} + \frac{4}{49} a^{5} + \frac{1}{7} a^{4} - \frac{8}{49} a^{2} - \frac{2}{49}$

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:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{40}{49} a^{11} + \frac{220}{49} a^{10} - \frac{650}{49} a^{9} + \frac{1275}{49} a^{8} - \frac{1818}{49} a^{7} + \frac{279}{7} a^{6} - \frac{1483}{49} a^{5} + \frac{100}{7} a^{4} - 3 a^{3} - \frac{23}{49} a^{2} - a + \frac{31}{49} \) (order $4$)
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:  \( 9.16575696339 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), 6.0.648000.1

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 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 ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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

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$2.12.12.26$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$$[2]^{6}$
$3$3.6.8.4$x^{6} + 18 x^{2} + 63$$3$$2$$8$$C_6$$[2]^{2}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.2e2_5.2t1.1c1$1$ $ 2^{2} \cdot 5 $ $x^{2} + 5$ $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.2e2_3e2_5.6t1.2c1$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{6} + 9 x^{4} - 2 x^{3} + 84 x^{2} + 36 x + 321$ $C_6$ (as 6T1) $0$ $-1$
1.3e2_5.6t1.1c1$1$ $ 3^{2} \cdot 5 $ $x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$ $C_6$ (as 6T1) $0$ $1$
1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.2e2_3e2.6t1.2c1$1$ $ 2^{2} \cdot 3^{2}$ $x^{6} + 6 x^{4} + 9 x^{2} + 1$ $C_6$ (as 6T1) $0$ $-1$
1.2e2_3e2_5.6t1.2c2$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{6} + 9 x^{4} - 2 x^{3} + 84 x^{2} + 36 x + 321$ $C_6$ (as 6T1) $0$ $-1$
1.2e2_3e2.6t1.2c2$1$ $ 2^{2} \cdot 3^{2}$ $x^{6} + 6 x^{4} + 9 x^{2} + 1$ $C_6$ (as 6T1) $0$ $-1$
1.3e2_5.6t1.1c2$1$ $ 3^{2} \cdot 5 $ $x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$ $C_6$ (as 6T1) $0$ $1$
2.2e2_3e4_5.3t2.2c1$2$ $ 2^{2} \cdot 3^{4} \cdot 5 $ $x^{3} - 3 x - 8$ $S_3$ (as 3T2) $1$ $0$
2.2e2_3e4_5.6t3.5c1$2$ $ 2^{2} \cdot 3^{4} \cdot 5 $ $x^{6} + 6 x^{4} + 9 x^{2} + 64$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e2_3e2_5.6t5.1c1$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{6} - 2 x^{5} + 2 x^{4} - 4 x^{3} + 7 x^{2} - 4 x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e2_3e2_5.12t18.1c1$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{12} - 6 x^{11} + 19 x^{10} - 40 x^{9} + 62 x^{8} - 74 x^{7} + 67 x^{6} - 44 x^{5} + 21 x^{4} - 8 x^{3} + 4 x^{2} - 2 x + 1$ $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.2e2_3e2_5.6t5.1c2$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{6} - 2 x^{5} + 2 x^{4} - 4 x^{3} + 7 x^{2} - 4 x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e2_3e2_5.12t18.1c2$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{12} - 6 x^{11} + 19 x^{10} - 40 x^{9} + 62 x^{8} - 74 x^{7} + 67 x^{6} - 44 x^{5} + 21 x^{4} - 8 x^{3} + 4 x^{2} - 2 x + 1$ $C_6\times S_3$ (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 *.