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)

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Normalized defining 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]);
 

\(x^{8} - 3 x^{7} + 7 x^{6} - 21 x^{5} + 42 x^{4} - 42 x^{3} + 28 x^{2} - 24 x + 16\)  Toggle raw display

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 \(a\))

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

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

\(\Q(\sqrt{-7}) \), 4.0.12348.1

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$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$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$$[\ ]_{4}^{2}$
$7$7.8.7.1$x^{8} + 14$$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 *.