# Properties

 Label 8.0.1429145856.1 Degree $8$ Signature $[0, 4]$ Discriminant $1429145856$ Root discriminant $$13.94$$ Ramified primes see page Class number $2$ Class group $[2]$ Galois group $D_{8}$ (as 8T6)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 5*x^6 - 2*x^5 + 21*x^4 - 30*x^3 + 75*x^2 - 62*x + 58)

gp: K = bnfinit(x^8 - 2*x^7 + 5*x^6 - 2*x^5 + 21*x^4 - 30*x^3 + 75*x^2 - 62*x + 58, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![58, -62, 75, -30, 21, -2, 5, -2, 1]);

$$x^{8} - 2x^{7} + 5x^{6} - 2x^{5} + 21x^{4} - 30x^{3} + 75x^{2} - 62x + 58$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 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: $$1429145856$$ 1429145856 $$\medspace = 2^{8}\cdot 3^{4}\cdot 41^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$13.94$$ 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: $$2$$, $$3$$, $$41$$ 2, 3, 41 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Aut(K/\Q) }$: $2$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{6}-\frac{1}{4}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{1888}a^{7}+\frac{65}{1888}a^{6}+\frac{7}{118}a^{5}-\frac{25}{944}a^{4}+\frac{447}{1888}a^{3}+\frac{655}{1888}a^{2}+\frac{2}{59}a-\frac{247}{944}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

 Monogenic: No Index: Not computed Inessential primes: $2$

## 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: $3$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-\frac{51}{1888} a^{7} - \frac{11}{1888} a^{6} - \frac{3}{118} a^{5} - \frac{141}{944} a^{4} - \frac{1085}{1888} a^{3} - \frac{837}{1888} a^{2} - \frac{27}{118} a - \frac{1091}{944}$$ -(51)/(1888)*a^(7) - (11)/(1888)*a^(6) - (3)/(118)*a^(5) - (141)/(944)*a^(4) - (1085)/(1888)*a^(3) - (837)/(1888)*a^(2) - (27)/(118)*a - (1091)/(944)  (order $4$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $\frac{97}{1888}a^{7}-\frac{67}{1888}a^{6}+\frac{1}{236}a^{5}+\frac{171}{944}a^{4}+\frac{1823}{1888}a^{3}-\frac{893}{1888}a^{2}+\frac{9}{236}a+\frac{1293}{944}$, $\frac{5}{944}a^{7}-\frac{265}{944}a^{6}+\frac{81}{236}a^{5}-\frac{243}{472}a^{4}-\frac{597}{944}a^{3}-\frac{4631}{944}a^{2}+\frac{1319}{236}a-\frac{3005}{472}$, $\frac{327}{1888}a^{7}+\frac{15}{1888}a^{6}+\frac{47}{118}a^{5}+\frac{793}{944}a^{4}+\frac{7401}{1888}a^{3}+\frac{3201}{1888}a^{2}+\frac{777}{118}a+\frac{5607}{944}$ 97/1888*a^7 - 67/1888*a^6 + 1/236*a^5 + 171/944*a^4 + 1823/1888*a^3 - 893/1888*a^2 + 9/236*a + 1293/944, 5/944*a^7 - 265/944*a^6 + 81/236*a^5 - 243/472*a^4 - 597/944*a^3 - 4631/944*a^2 + 1319/236*a - 3005/472, 327/1888*a^7 + 15/1888*a^6 + 47/118*a^5 + 793/944*a^4 + 7401/1888*a^3 + 3201/1888*a^2 + 777/118*a + 5607/944 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$152.32463607$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 152.32463607 \cdot 2}{4\sqrt{1429145856}}\approx 3.1399400030$

## Galois group

$D_8$ (as 8T6):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 16 The 7 conjugacy class representatives for $D_{8}$ Character table for $D_{8}$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Sibling fields

 Galois closure: 16.0.3433371692450636169216.3 Degree 8 sibling: 8.2.14648745024.1

## 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 ${\href{/padicField/5.4.0.1}{4} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }$ ${\href{/padicField/11.8.0.1}{8} }$ ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }$ ${\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ R ${\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }$ ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$$2$$ 2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2] 2.2.2.1x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2] 2.2.2.1x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$$3$$ 3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4} $$41$$ 41.2.0.1x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2} 41.4.2.1x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $2^{2}$ $$\Q(\sqrt{-1})$$ $C_2$ (as 2T1) $1$ $-1$
1.41.2t1.a.a$1$ $41$ $$\Q(\sqrt{41})$$ $C_2$ (as 2T1) $1$ $1$
1.164.2t1.a.a$1$ $2^{2} \cdot 41$ $$\Q(\sqrt{-41})$$ $C_2$ (as 2T1) $1$ $-1$
* 2.164.4t3.a.a$2$ $2^{2} \cdot 41$ 4.2.6724.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.1476.8t6.a.a$2$ $2^{2} \cdot 3^{2} \cdot 41$ 8.0.1429145856.1 $D_{8}$ (as 8T6) $1$ $0$
* 2.1476.8t6.a.b$2$ $2^{2} \cdot 3^{2} \cdot 41$ 8.0.1429145856.1 $D_{8}$ (as 8T6) $1$ $0$

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 *.