# Properties

 Label 12.0.29365647704064.2 Degree $12$ Signature $[0, 6]$ Discriminant $2.937\times 10^{13}$ Root discriminant $$13.25$$ Ramified primes see page 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 - 4*x^11 + 10*x^10 - 16*x^9 + 20*x^8 - 20*x^7 + 18*x^6 - 20*x^5 + 7*x^4 + 4*x^3 + 4*x^2 - 4*x + 1)

gp: K = bnfinit(x^12 - 4*x^11 + 10*x^10 - 16*x^9 + 20*x^8 - 20*x^7 + 18*x^6 - 20*x^5 + 7*x^4 + 4*x^3 + 4*x^2 - 4*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 4, 4, 7, -20, 18, -20, 20, -16, 10, -4, 1]);

$$x^{12} - 4 x^{11} + 10 x^{10} - 16 x^{9} + 20 x^{8} - 20 x^{7} + 18 x^{6} - 20 x^{5} + 7 x^{4} + 4 x^{3} + 4 x^{2} - 4 x + 1$$

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: $$29365647704064$$ 29365647704064 $$\medspace = 2^{24}\cdot 3^{6}\cdot 7^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$13.25$$ 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: $$2$$, $$3$$, $$7$$ 2, 3, 7 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{15301}a^{11}-\frac{6605}{15301}a^{10}+\frac{7066}{15301}a^{9}-\frac{5234}{15301}a^{8}-\frac{4}{15301}a^{7}-\frac{4218}{15301}a^{6}-\frac{368}{1177}a^{5}-\frac{2100}{15301}a^{4}-\frac{599}{15301}a^{3}+\frac{6345}{15301}a^{2}-\frac{4504}{15301}a+\frac{1057}{15301}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

 Monogenic: Not computed Index: $1$ Inessential primes: None

## 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: $$-1$$ -1  (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $\frac{1181}{15301}a^{11}+\frac{3005}{15301}a^{10}-\frac{9400}{15301}a^{9}+\frac{30852}{15301}a^{8}-\frac{35326}{15301}a^{7}+\frac{52571}{15301}a^{6}-\frac{2649}{1177}a^{5}+\frac{44565}{15301}a^{4}-\frac{64777}{15301}a^{3}-\frac{34647}{15301}a^{2}+\frac{5524}{15301}a+\frac{8936}{15301}$, $\frac{12711}{15301}a^{11}-\frac{45471}{15301}a^{10}+\frac{106163}{15301}a^{9}-\frac{153636}{15301}a^{8}+\frac{178671}{15301}a^{7}-\frac{168605}{15301}a^{6}+\frac{11520}{1177}a^{5}-\frac{191768}{15301}a^{4}+\frac{6009}{15301}a^{3}+\frac{60928}{15301}a^{2}+\frac{82503}{15301}a-\frac{14052}{15301}$, $\frac{2594}{15301}a^{11}-\frac{11551}{15301}a^{10}+\frac{29208}{15301}a^{9}-\frac{50912}{15301}a^{8}+\frac{66129}{15301}a^{7}-\frac{77782}{15301}a^{6}+\frac{5840}{1177}a^{5}-\frac{92050}{15301}a^{4}+\frac{52799}{15301}a^{3}-\frac{4946}{15301}a^{2}+\frac{21889}{15301}a-\frac{12322}{15301}$, $\frac{10117}{15301}a^{11}-\frac{33920}{15301}a^{10}+\frac{76955}{15301}a^{9}-\frac{102724}{15301}a^{8}+\frac{112542}{15301}a^{7}-\frac{90823}{15301}a^{6}+\frac{5680}{1177}a^{5}-\frac{99718}{15301}a^{4}-\frac{46790}{15301}a^{3}+\frac{65874}{15301}a^{2}+\frac{60614}{15301}a-\frac{17031}{15301}$, $\frac{8936}{15301}a^{11}-\frac{36925}{15301}a^{10}+\frac{86355}{15301}a^{9}-\frac{133576}{15301}a^{8}+\frac{147868}{15301}a^{7}-\frac{143394}{15301}a^{6}+\frac{8329}{1177}a^{5}-\frac{144283}{15301}a^{4}+\frac{17987}{15301}a^{3}+\frac{100521}{15301}a^{2}+\frac{70391}{15301}a-\frac{25967}{15301}$ 1181/15301*a^11 + 3005/15301*a^10 - 9400/15301*a^9 + 30852/15301*a^8 - 35326/15301*a^7 + 52571/15301*a^6 - 2649/1177*a^5 + 44565/15301*a^4 - 64777/15301*a^3 - 34647/15301*a^2 + 5524/15301*a + 8936/15301, 12711/15301*a^11 - 45471/15301*a^10 + 106163/15301*a^9 - 153636/15301*a^8 + 178671/15301*a^7 - 168605/15301*a^6 + 11520/1177*a^5 - 191768/15301*a^4 + 6009/15301*a^3 + 60928/15301*a^2 + 82503/15301*a - 14052/15301, 2594/15301*a^11 - 11551/15301*a^10 + 29208/15301*a^9 - 50912/15301*a^8 + 66129/15301*a^7 - 77782/15301*a^6 + 5840/1177*a^5 - 92050/15301*a^4 + 52799/15301*a^3 - 4946/15301*a^2 + 21889/15301*a - 12322/15301, 10117/15301*a^11 - 33920/15301*a^10 + 76955/15301*a^9 - 102724/15301*a^8 + 112542/15301*a^7 - 90823/15301*a^6 + 5680/1177*a^5 - 99718/15301*a^4 - 46790/15301*a^3 + 65874/15301*a^2 + 60614/15301*a - 17031/15301, 8936/15301*a^11 - 36925/15301*a^10 + 86355/15301*a^9 - 133576/15301*a^8 + 147868/15301*a^7 - 143394/15301*a^6 + 8329/1177*a^5 - 144283/15301*a^4 + 17987/15301*a^3 + 100521/15301*a^2 + 70391/15301*a - 25967/15301 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$29.848139323999433$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 29.848139323999433 \cdot 2}{2\sqrt{29365647704064}}\approx 0.338903989446607$

## 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: 18.0.86675003008018355847168.2, 18.6.292528135152061950984192.1

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R R ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ R ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{6}$ ${\href{/padicField/13.2.0.1}{2} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

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
$$2$$ 2.12.24.307$x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3} $$3$$ 3.6.3.1x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3} $$7$$ 7.6.0.1x^{6} + 3 x^{2} - x + 5$$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$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.8.2t1.b.a$1$ $2^{3}$ $$\Q(\sqrt{-2})$$ $C_2$ (as 2T1) $1$ $-1$
* 1.24.2t1.b.a$1$ $2^{3} \cdot 3$ $$\Q(\sqrt{-6})$$ $C_2$ (as 2T1) $1$ $-1$
* 1.12.2t1.a.a$1$ $2^{2} \cdot 3$ $$\Q(\sqrt{3})$$ $C_2$ (as 2T1) $1$ $1$
1.84.6t1.a.a$1$ $2^{2} \cdot 3 \cdot 7$ 6.6.4148928.1 $C_6$ (as 6T1) $0$ $1$
1.56.6t1.c.a$1$ $2^{3} \cdot 7$ 6.0.1229312.1 $C_6$ (as 6T1) $0$ $-1$
1.56.6t1.c.b$1$ $2^{3} \cdot 7$ 6.0.1229312.1 $C_6$ (as 6T1) $0$ $-1$
1.168.6t1.c.a$1$ $2^{3} \cdot 3 \cdot 7$ 6.0.33191424.1 $C_6$ (as 6T1) $0$ $-1$
1.84.6t1.a.b$1$ $2^{2} \cdot 3 \cdot 7$ 6.6.4148928.1 $C_6$ (as 6T1) $0$ $1$
1.168.6t1.c.b$1$ $2^{3} \cdot 3 \cdot 7$ 6.0.33191424.1 $C_6$ (as 6T1) $0$ $-1$
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) $0$ $1$
2.1176.3t2.a.a$2$ $2^{3} \cdot 3 \cdot 7^{2}$ 3.1.1176.1 $S_3$ (as 3T2) $1$ $0$
2.4704.6t3.f.a$2$ $2^{5} \cdot 3 \cdot 7^{2}$ 6.0.44255232.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.168.6t5.c.a$2$ $2^{3} \cdot 3 \cdot 7$ 6.0.677376.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.168.6t5.c.b$2$ $2^{3} \cdot 3 \cdot 7$ 6.0.677376.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.672.12t18.c.a$2$ $2^{5} \cdot 3 \cdot 7$ 12.0.29365647704064.2 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.672.12t18.c.b$2$ $2^{5} \cdot 3 \cdot 7$ 12.0.29365647704064.2 $C_6\times S_3$ (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 *.