# Properties

 Label 12.0.205924456521.1 Degree $12$ Signature $[0, 6]$ Discriminant $205924456521$ Root discriminant $8.77$ Ramified primes $3, 7$ Class number $1$ Class group trivial Galois group $C_6\times C_2$ (as 12T2)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1)

gp: K = bnfinit(x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1, 1)

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

$$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

 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: $$205924456521$$$$\medspace = 3^{6}\cdot 7^{10}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $8.77$ 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: $3, 7$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $12$ This field is Galois and abelian over $\Q$. Conductor: $$21=3\cdot 7$$ Dirichlet character group: $\lbrace$$\chi_{21}(1,·), \chi_{21}(2,·), \chi_{21}(4,·), \chi_{21}(5,·), \chi_{21}(8,·), \chi_{21}(10,·), \chi_{21}(11,·), \chi_{21}(13,·), \chi_{21}(16,·), \chi_{21}(17,·), \chi_{21}(19,·)$$\chi_{21}(20,·)$$\rbrace This is a CM field. ## Integral basis (with respect to field generator $$a$$) 1, a, a^{2}, a^{3}, a^{4}, a^{5}, a^{6}, a^{7}, a^{8}, a^{9}, a^{10}, a^{11} sage: K.integral_basis() gp: K.zk magma: IntegralBasis(K); ## Class group and class number Trivial group, which has order 1 sage: K.class_group().invariants() gp: K.clgp magma: ClassGroup(K); ## Unit group sage: UK = K.unit_group() magma: UK, f := UnitGroup(K);  Rank: 5 sage: UK.rank() gp: K.fu magma: UnitRank(K); Torsion generator: $$-a$$ (order 42) sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$a^{3} + 1$$, $$a^{10} - a^{8} + a^{3}$$, $$a - 1$$, $$a^{2} - 1$$, $$a^{4} - 1$$ sage: UK.fundamental_units() gp: K.fu magma: [K!f(g): g in Generators(UK)]; Regulator: $$70.3993980027$$ 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 70.3993980027 \cdot 1}{42\sqrt{205924456521}}\approx 0.227271459514 ## Galois group C_2\times C_6 (as 12T2): sage: K.galois_group(type='pari') gp: polgalois(K.pol) magma: GaloisGroup(K);  An abelian group of order 12 The 12 conjugacy class representatives for C_6\times C_2 Character table for C_6\times C_2 ## Intermediate fields Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities. ## 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} }^{2} R {\href{/LocalNumberField/5.6.0.1}{6} }^{2} R {\href{/LocalNumberField/11.6.0.1}{6} }^{2} {\href{/LocalNumberField/13.2.0.1}{2} }^{6} {\href{/LocalNumberField/17.6.0.1}{6} }^{2} {\href{/LocalNumberField/19.6.0.1}{6} }^{2} {\href{/LocalNumberField/23.6.0.1}{6} }^{2} {\href{/LocalNumberField/29.2.0.1}{2} }^{6} {\href{/LocalNumberField/31.6.0.1}{6} }^{2} {\href{/LocalNumberField/37.3.0.1}{3} }^{4} {\href{/LocalNumberField/41.2.0.1}{2} }^{6} {\href{/LocalNumberField/43.1.0.1}{1} }^{12} {\href{/LocalNumberField/47.6.0.1}{6} }^{2} {\href{/LocalNumberField/53.6.0.1}{6} }^{2} {\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. 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 33.12.6.2x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6} 7.6.5.5x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$