# Properties

 Label 6.0.50602347.1 Degree $6$ Signature $[0, 3]$ Discriminant $-50602347$ Root discriminant $19.23$ Ramified primes $3, 37$ Class number $3$ Class group $[3]$ Galois group $C_6$ (as 6T1)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 13*x^4 + 34*x^3 + 133*x^2 + 132*x + 121)

gp: K = bnfinit(x^6 - x^5 + 13*x^4 + 34*x^3 + 133*x^2 + 132*x + 121, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, 132, 133, 34, 13, -1, 1]);

$$x^{6} - x^{5} + 13 x^{4} + 34 x^{3} + 133 x^{2} + 132 x + 121$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 3]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-50602347$$$$\medspace = -\,3^{3}\cdot 37^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $19.23$ 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: $3, 37$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $6$ This field is Galois and abelian over $\Q$. Conductor: $$111=3\cdot 37$$ Dirichlet character group: $\lbrace$$\chi_{111}(1,·), \chi_{111}(100,·), \chi_{111}(38,·), \chi_{111}(10,·), \chi_{111}(26,·)$$\chi_{111}(47,·)$$\rbrace This is a CM field. ## Integral basis (with respect to field generator $$a$$) 1, a, a^{2}, a^{3}, a^{4}, \frac{1}{20471} a^{5} - \frac{9318}{20471} a^{4} - \frac{1692}{20471} a^{3} + \frac{1728}{20471} a^{2} - \frac{9437}{20471} a + \frac{156}{1861} sage: K.integral_basis() gp: K.zk magma: IntegralBasis(K); ## Class group and class number C_{3}, which has order 3 sage: K.class_group().invariants() gp: K.clgp magma: ClassGroup(K); ## Unit group sage: UK = K.unit_group() magma: UK, f := UnitGroup(K);  Rank: 2 sage: UK.rank() gp: K.fu magma: UnitRank(K); Torsion generator: $$\frac{156}{20471} a^{5} - \frac{167}{20471} a^{4} + \frac{2171}{20471} a^{3} + \frac{3445}{20471} a^{2} + \frac{22211}{20471} a + \frac{2004}{1861}$$ (order 6) sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$\frac{145}{20471} a^{5} - \frac{24}{20471} a^{4} + \frac{312}{20471} a^{3} + \frac{4908}{20471} a^{2} + \frac{3192}{20471} a + \frac{288}{1861}$$, $$\frac{301}{20471} a^{5} - \frac{191}{20471} a^{4} + \frac{2483}{20471} a^{3} + \frac{8353}{20471} a^{2} + \frac{25403}{20471} a + \frac{2292}{1861}$$ sage: UK.fundamental_units() gp: K.fu magma: [K!f(g): g in Generators(UK)]; Regulator: $$12.5049207637$$ sage: K.regulator() gp: K.reg magma: Regulator(K); ## Class number formula \displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 12.5049207637 \cdot 3}{6\sqrt{50602347}}\approx 0.218024461956 ## Galois group C_6 (as 6T1): sage: K.galois_group(type='pari') gp: polgalois(K.pol) magma: GaloisGroup(K);  A cyclic group of order 6 The 6 conjugacy class representatives for C_6 Character table for C_6 ## Intermediate fields Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities. ## Sibling algebras  Twin sextic algebra: 3.3.1369.1 \times $$\Q(\sqrt{-3})$$ \times $$\Q$$ ## Frobenius cycle types  p 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 Cycle type {\href{/LocalNumberField/2.6.0.1}{6} } R {\href{/LocalNumberField/5.6.0.1}{6} } {\href{/LocalNumberField/7.3.0.1}{3} }^{2} {\href{/LocalNumberField/11.2.0.1}{2} }^{3} {\href{/LocalNumberField/13.3.0.1}{3} }^{2} {\href{/LocalNumberField/17.6.0.1}{6} } {\href{/LocalNumberField/19.3.0.1}{3} }^{2} {\href{/LocalNumberField/23.2.0.1}{2} }^{3} {\href{/LocalNumberField/29.2.0.1}{2} }^{3} {\href{/LocalNumberField/31.1.0.1}{1} }^{6} R {\href{/LocalNumberField/41.6.0.1}{6} } {\href{/LocalNumberField/43.1.0.1}{1} }^{6} {\href{/LocalNumberField/47.2.0.1}{2} }^{3} {\href{/LocalNumberField/53.6.0.1}{6} } {\href{/LocalNumberField/59.6.0.1}{6} } In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents. sage: p = 7; # to obtain a list of [e_i,f_i] for the factorization of the ideal p\mathcal{O}_K: sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)] gp: p = 7; \\ to obtain a list of [e_i,f_i] for the factorization of the ideal p\mathcal{O}_K: gp: idealfactors = idealprimedec(K, p); \\ get the data gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]]) magma: p := 7; // to obtain a list of [e_i,f_i] for the factorization of the ideal p\mathcal{O}_K: magma: idealfactors := Factorization(p*Integers(K)); // get the data magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors]; ## Local algebras for ramified primes pLabelPolynomial e f c Galois group Slope content 33.6.3.2x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3} 37.3.2.1x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $3$ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.37.3t1.a.a$1$ $37$ $x^{3} - x^{2} - 12 x - 11$ $C_3$ (as 3T1) $0$ $1$
* 1.111.6t1.b.a$1$ $3 \cdot 37$ $x^{6} - x^{5} + 13 x^{4} + 34 x^{3} + 133 x^{2} + 132 x + 121$ $C_6$ (as 6T1) $0$ $-1$
* 1.37.3t1.a.b$1$ $37$ $x^{3} - x^{2} - 12 x - 11$ $C_3$ (as 3T1) $0$ $1$
* 1.111.6t1.b.b$1$ $3 \cdot 37$ $x^{6} - x^{5} + 13 x^{4} + 34 x^{3} + 133 x^{2} + 132 x + 121$ $C_6$ (as 6T1) $0$ $-1$

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 *.