# Properties

 Label 8.0.38423222208.1 Degree $8$ Signature $[0, 4]$ Discriminant $38423222208$ Root discriminant $21.04$ Ramified primes $2, 3, 7$ Class number $4$ Class group $[4]$ 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 - 3*x^7 + 7*x^6 - 21*x^5 + 42*x^4 - 42*x^3 + 28*x^2 - 24*x + 16)

gp: K = bnfinit(x^8 - 3*x^7 + 7*x^6 - 21*x^5 + 42*x^4 - 42*x^3 + 28*x^2 - 24*x + 16, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -24, 28, -42, 42, -21, 7, -3, 1]);

$$x^{8} - 3 x^{7} + 7 x^{6} - 21 x^{5} + 42 x^{4} - 42 x^{3} + 28 x^{2} - 24 x + 16$$

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: $$38423222208$$$$\medspace = 2^{6}\cdot 3^{6}\cdot 7^{7}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $21.04$ 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: $2, 3, 7$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{3}{16} a^{5} - \frac{3}{16} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{4}$, which has order $4$

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: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$\frac{1}{8} a^{7} - \frac{3}{8} a^{6} + \frac{3}{8} a^{5} - \frac{17}{8} a^{4} + \frac{15}{4} a^{3} + \frac{5}{4} a^{2} - \frac{1}{2} a - 3$$,  $$\frac{3}{8} a^{7} - \frac{9}{8} a^{6} + \frac{17}{8} a^{5} - \frac{59}{8} a^{4} + \frac{57}{4} a^{3} - \frac{37}{4} a^{2} + \frac{9}{2} a - 9$$,  $$\frac{13}{8} a^{7} - \frac{23}{8} a^{6} + \frac{55}{8} a^{5} - \frac{205}{8} a^{4} + \frac{135}{4} a^{3} - \frac{79}{4} a^{2} + \frac{33}{2} a - 17$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$476.992507974$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 476.992507974 \cdot 4}{2\sqrt{38423222208}}\approx 7.58514933670$

## 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: Deg 16 Degree 8 sibling: 8.2.153692888832.3

## 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 ${\href{/padicField/5.8.0.1}{8} }$ R ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }$ ${\href{/padicField/17.8.0.1}{8} }$ ${\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }$ ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{4}$

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$$\Q_{2}$$x + 1$$1$$1$$0Trivial[\ ] \Q_{2}$$x + 1$$1$$1$$0Trivial[\ ] 2.2.2.1x^{2} + 2 x + 2$$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$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2} 77.8.7.1x^{8} + 14$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
1.28.2t1.a.a$1$ $2^{2} \cdot 7$ $$\Q(\sqrt{7})$$ $C_2$ (as 2T1) $1$ $1$
* 1.7.2t1.a.a$1$ $7$ $$\Q(\sqrt{-7})$$ $C_2$ (as 2T1) $1$ $-1$
1.4.2t1.a.a$1$ $2^{2}$ $$\Q(\sqrt{-1})$$ $C_2$ (as 2T1) $1$ $-1$
* 2.1764.4t3.c.a$2$ $2^{2} \cdot 3^{2} \cdot 7^{2}$ 4.2.49392.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.1764.8t6.a.a$2$ $2^{2} \cdot 3^{2} \cdot 7^{2}$ 8.0.38423222208.1 $D_{8}$ (as 8T6) $1$ $0$
* 2.1764.8t6.a.b$2$ $2^{2} \cdot 3^{2} \cdot 7^{2}$ 8.0.38423222208.1 $D_{8}$ (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 *.