# Properties

 Label 12.0.153664000000.1 Degree $12$ Signature $[0, 6]$ Discriminant $153664000000$ Root discriminant $8.55$ Ramified primes $2, 5, 7$ Class number $1$ Class group trivial Galois group $C_6\times S_3$ (as 12T18)

# Learn more

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 + x^10 - 4*x^9 + 4*x^8 - 6*x^7 + 12*x^6 - 12*x^5 + 12*x^4 - 16*x^3 + 14*x^2 - 6*x + 1)

gp: K = bnfinit(x^12 + x^10 - 4*x^9 + 4*x^8 - 6*x^7 + 12*x^6 - 12*x^5 + 12*x^4 - 16*x^3 + 14*x^2 - 6*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 14, -16, 12, -12, 12, -6, 4, -4, 1, 0, 1]);

$$x^{12} + x^{10} - 4 x^{9} + 4 x^{8} - 6 x^{7} + 12 x^{6} - 12 x^{5} + 12 x^{4} - 16 x^{3} + 14 x^{2} - 6 x + 1$$

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: $$153664000000$$$$\medspace = 2^{12}\cdot 5^{6}\cdot 7^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $8.55$ 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: $2, 5, 7$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $6$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## 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: $$\frac{14}{3} a^{11} + 3 a^{10} + 7 a^{9} - \frac{41}{3} a^{8} + \frac{32}{3} a^{7} - \frac{67}{3} a^{6} + \frac{124}{3} a^{5} - \frac{95}{3} a^{4} + \frac{116}{3} a^{3} - 51 a^{2} + \frac{107}{3} a - 9$$ (order $4$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); 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 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$5.58654010372$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 5.58654010372 \cdot 1}{4\sqrt{153664000000}}\approx 0.219217930008$

## 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

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: 18.6.110730297608000000000.1, 18.0.56693912375296000000.1

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{3}$ R R ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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
$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$$[2]^{6} 55.6.3.1x^{6} - 10 x^{4} + 25 x^{2} - 500$$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$$[\ ]_{2}^{3} 77.6.0.1x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $2^{2}$ $$\Q(\sqrt{-1})$$ $C_2$ (as 2T1) $1$ $-1$
* 1.20.2t1.a.a$1$ $2^{2} \cdot 5$ $$\Q(\sqrt{-5})$$ $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $5$ $$\Q(\sqrt{5})$$ $C_2$ (as 2T1) $1$ $1$
1.7.3t1.a.a$1$ $7$ $$\Q(\zeta_{7})^+$$ $C_3$ (as 3T1) $0$ $1$
1.7.3t1.a.b$1$ $7$ $$\Q(\zeta_{7})^+$$ $C_3$ (as 3T1) $0$ $1$
1.140.6t1.b.a$1$ $2^{2} \cdot 5 \cdot 7$ 6.0.19208000.1 $C_6$ (as 6T1) $0$ $-1$
1.140.6t1.b.b$1$ $2^{2} \cdot 5 \cdot 7$ 6.0.19208000.1 $C_6$ (as 6T1) $0$ $-1$
1.35.6t1.b.a$1$ $5 \cdot 7$ 6.6.300125.1 $C_6$ (as 6T1) $0$ $1$
1.28.6t1.a.a$1$ $2^{2} \cdot 7$ 6.0.153664.1 $C_6$ (as 6T1) $0$ $-1$
1.28.6t1.a.b$1$ $2^{2} \cdot 7$ 6.0.153664.1 $C_6$ (as 6T1) $0$ $-1$
1.35.6t1.b.b$1$ $5 \cdot 7$ 6.6.300125.1 $C_6$ (as 6T1) $0$ $1$
2.980.3t2.a.a$2$ $2^{2} \cdot 5 \cdot 7^{2}$ 3.1.980.1 $S_3$ (as 3T2) $1$ $0$
2.980.6t3.e.a$2$ $2^{2} \cdot 5 \cdot 7^{2}$ 6.0.3841600.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.140.12t18.a.a$2$ $2^{2} \cdot 5 \cdot 7$ 12.0.153664000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.140.6t5.a.a$2$ $2^{2} \cdot 5 \cdot 7$ 6.0.392000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.140.12t18.a.b$2$ $2^{2} \cdot 5 \cdot 7$ 12.0.153664000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.140.6t5.a.b$2$ $2^{2} \cdot 5 \cdot 7$ 6.0.392000.1 $S_3\times C_3$ (as 6T5) $0$ $0$

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 *.