# Properties

 Label 12.0.181612683140625.1 Degree $12$ Signature $[0, 6]$ Discriminant $1.816\times 10^{14}$ Root discriminant $15.43$ Ramified primes $3, 5, 11$ Class number $2$ Class group $[2]$ Galois group $C_6\times S_3$ (as 12T18)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^10 - 8*x^9 + 22*x^8 + 13*x^7 + 5*x^6 + 2*x^5 + 8*x^4 + 11*x^3 + 8*x^2 - x + 1)

gp: K = bnfinit(x^12 - 3*x^10 - 8*x^9 + 22*x^8 + 13*x^7 + 5*x^6 + 2*x^5 + 8*x^4 + 11*x^3 + 8*x^2 - x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 8, 11, 8, 2, 5, 13, 22, -8, -3, 0, 1]);

$$x^{12} - 3 x^{10} - 8 x^{9} + 22 x^{8} + 13 x^{7} + 5 x^{6} + 2 x^{5} + 8 x^{4} + 11 x^{3} + 8 x^{2} - x + 1$$

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: $$181612683140625$$$$\medspace = 3^{8}\cdot 5^{6}\cdot 11^{6}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $15.43$ 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: $3, 5, 11$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{1}{10} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{2345150} a^{11} - \frac{8581}{2345150} a^{10} + \frac{230363}{2345150} a^{9} - \frac{721521}{2345150} a^{8} + \frac{175723}{2345150} a^{7} + \frac{151189}{469030} a^{6} - \frac{54817}{234515} a^{5} - \frac{27549}{1172575} a^{4} + \frac{5347}{13175} a^{3} + \frac{23071}{469030} a^{2} - \frac{1146007}{2345150} a + \frac{437601}{2345150}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}$, which has order $2$

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: $$-1$$ (order $2$) 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: $$86.95395829502839$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 86.95395829502839 \cdot 2}{2\sqrt{181612683140625}}\approx 0.397004545657124$

## 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: 18.6.977228812651986017578125.1, 18.0.1300691549639793389396484375.1

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}$ R R ${\href{/padicField/7.6.0.1}{6} }^{2}$ R ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{6}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

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$3.6.8.4$x^{6} + 18 x^{2} + 63$$3$$2$$8$$C_6$$[2]^{2} 3.6.0.1x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2} 5.2.1.1x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2} 5.6.3.1x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3} 11.6.3.2x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.55.2t1.a.a$1$ $5 \cdot 11$ $$\Q(\sqrt{-55})$$ $C_2$ (as 2T1) $1$ $-1$
* 1.11.2t1.a.a$1$ $11$ $$\Q(\sqrt{-11})$$ $C_2$ (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) $0$ $1$
1.495.6t1.b.a$1$ $3^{2} \cdot 5 \cdot 11$ 6.0.1091586375.3 $C_6$ (as 6T1) $0$ $-1$
1.495.6t1.b.b$1$ $3^{2} \cdot 5 \cdot 11$ 6.0.1091586375.3 $C_6$ (as 6T1) $0$ $-1$
1.99.6t1.a.a$1$ $3^{2} \cdot 11$ 6.0.8732691.1 $C_6$ (as 6T1) $0$ $-1$
1.45.6t1.a.b$1$ $3^{2} \cdot 5$ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
1.9.3t1.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) $0$ $1$
1.99.6t1.a.b$1$ $3^{2} \cdot 11$ 6.0.8732691.1 $C_6$ (as 6T1) $0$ $-1$
2.891.3t2.b.a$2$ $3^{4} \cdot 11$ 3.1.891.1 $S_3$ (as 3T2) $1$ $0$
2.22275.6t3.c.a$2$ $3^{4} \cdot 5^{2} \cdot 11$ 6.0.1091586375.2 $D_{6}$ (as 6T3) $1$ $0$
* 2.99.6t5.a.a$2$ $3^{2} \cdot 11$ 6.0.107811.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2475.12t18.b.a$2$ $3^{2} \cdot 5^{2} \cdot 11$ 12.0.181612683140625.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.2475.12t18.b.b$2$ $3^{2} \cdot 5^{2} \cdot 11$ 12.0.181612683140625.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.99.6t5.a.b$2$ $3^{2} \cdot 11$ 6.0.107811.1 $S_3\times C_3$ (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 *.