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)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining 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]);
 

\( x^{8} - 2x^{7} + 5x^{6} - 2x^{5} + 21x^{4} - 30x^{3} + 75x^{2} - 62x + 58 \) Copy content 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:   \(1429145856\) \(\medspace = 2^{8}\cdot 3^{4}\cdot 41^{3}\) Copy content Toggle raw display
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:   \(2\), \(3\), \(41\) Copy content Toggle raw display
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 \(a\))

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

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:   \( -\frac{51}{1888} a^{7} - \frac{11}{1888} a^{6} - \frac{3}{118} a^{5} - \frac{141}{944} a^{4} - \frac{1085}{1888} a^{3} - \frac{837}{1888} a^{2} - \frac{27}{118} a - \frac{1091}{944} \)  (order $4$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $\frac{97}{1888}a^{7}-\frac{67}{1888}a^{6}+\frac{1}{236}a^{5}+\frac{171}{944}a^{4}+\frac{1823}{1888}a^{3}-\frac{893}{1888}a^{2}+\frac{9}{236}a+\frac{1293}{944}$, $\frac{5}{944}a^{7}-\frac{265}{944}a^{6}+\frac{81}{236}a^{5}-\frac{243}{472}a^{4}-\frac{597}{944}a^{3}-\frac{4631}{944}a^{2}+\frac{1319}{236}a-\frac{3005}{472}$, $\frac{327}{1888}a^{7}+\frac{15}{1888}a^{6}+\frac{47}{118}a^{5}+\frac{793}{944}a^{4}+\frac{7401}{1888}a^{3}+\frac{3201}{1888}a^{2}+\frac{777}{118}a+\frac{5607}{944}$ Copy content Toggle raw display
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

\(\Q(\sqrt{-1}) \), 4.0.656.1

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
\(2\) Copy content Toggle raw display 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$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
\(3\) Copy content Toggle raw display 3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
\(41\) Copy content Toggle raw display 41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.2.1$x^{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 *.