# Properties

 Label 8.0.1265625.1 Degree $8$ Signature $[0, 4]$ Discriminant $1265625$ Root discriminant $5.79$ Ramified primes $3, 5$ Class number $1$ Class group trivial Galois group $C_4\times C_2$ (as 8T2)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)

gp: K = bnfinit(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1, 1)

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

$$x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 4]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$1265625$$$$\medspace = 3^{4}\cdot 5^{6}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $5.79$ 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: $3, 5$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $8$ This field is Galois and abelian over $\Q$. Conductor: $$15=3\cdot 5$$ Dirichlet character group: $\lbrace$$\chi_{15}(1,·), \chi_{15}(2,·), \chi_{15}(4,·), \chi_{15}(7,·), \chi_{15}(8,·), \chi_{15}(11,·), \chi_{15}(13,·)$$\chi_{15}(14,·)$$\rbrace This is a CM field. ## Integral basis (with respect to field generator $$a$$) 1, a, a^{2}, a^{3}, a^{4}, a^{5}, a^{6}, a^{7} sage: K.integral_basis() gp: K.zk magma: IntegralBasis(K); ## Class group and class number Trivial group, which has order 1 sage: K.class_group().invariants() gp: K.clgp magma: ClassGroup(K); ## Unit group sage: UK = K.unit_group() magma: UK, f := UnitGroup(K);  Rank: 3 sage: UK.rank() gp: K.fu magma: UnitRank(K); Torsion generator: $$-a$$ (order 30) sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$a^{3} + 1$$, $$a - 1$$, $$a^{2} - 1$$ sage: UK.fundamental_units() gp: K.fu magma: [K!f(g): g in Generators(UK)]; Regulator: $$4.6618207773$$ 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)^{4}\cdot 4.6618207773 \cdot 1}{30\sqrt{1265625}}\approx 0.21527880272 ## Galois group C_2\times C_4 (as 8T2): sage: K.galois_group(type='pari') gp: polgalois(K.pol) magma: GaloisGroup(K);  An abelian group of order 8 The 8 conjugacy class representatives for C_4\times C_2 Character table for C_4\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/2.4.0.1}{4} }^{2} R R {\href{/LocalNumberField/7.4.0.1}{4} }^{2} {\href{/LocalNumberField/11.2.0.1}{2} }^{4} {\href{/LocalNumberField/13.4.0.1}{4} }^{2} {\href{/LocalNumberField/17.4.0.1}{4} }^{2} {\href{/LocalNumberField/19.2.0.1}{2} }^{4} {\href{/LocalNumberField/23.4.0.1}{4} }^{2} {\href{/LocalNumberField/29.2.0.1}{2} }^{4} {\href{/LocalNumberField/31.1.0.1}{1} }^{8} {\href{/LocalNumberField/37.4.0.1}{4} }^{2} {\href{/LocalNumberField/41.2.0.1}{2} }^{4} {\href{/LocalNumberField/43.4.0.1}{4} }^{2} {\href{/LocalNumberField/47.4.0.1}{4} }^{2} {\href{/LocalNumberField/53.4.0.1}{4} }^{2} {\href{/LocalNumberField/59.2.0.1}{2} }^{4} 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 33.8.4.1x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.15.2t1.a.a$1$ $3 \cdot 5$ $x^{2} - x + 4$ $C_2$ (as 2T1) $1$ $-1$
* 1.15.4t1.a.a$1$ $3 \cdot 5$ $x^{4} - x^{3} - 4 x^{2} + 4 x + 1$ $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.a.a$1$ $5$ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 1.5.2t1.a.a$1$ $5$ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.3.2t1.a.a$1$ $3$ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.15.4t1.a.b$1$ $3 \cdot 5$ $x^{4} - x^{3} - 4 x^{2} + 4 x + 1$ $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.a.b$1$ $5$ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$

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 *.