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Properties

Label 8.2.8671400783.1
Degree $8$
Signature $[2, 3]$
Discriminant $-8671400783$
Root discriminant \(17.47\)
Ramified primes see page
Class number $1$
Class group trivial
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 - x^7 + 2*x^6 + x^5 - 15*x^4 + 23*x^3 + 6*x^2 - 27*x + 19)
 
gp: K = bnfinit(x^8 - x^7 + 2*x^6 + x^5 - 15*x^4 + 23*x^3 + 6*x^2 - 27*x + 19, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, -27, 6, 23, -15, 1, 2, -1, 1]);
 

\( x^{8} - x^{7} + 2x^{6} + x^{5} - 15x^{4} + 23x^{3} + 6x^{2} - 27x + 19 \) 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:  $[2, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(-8671400783\) \(\medspace = -\,17^{4}\cdot 47^{3}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(17.47\)
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:   \(17\), \(47\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{273}a^{7}-\frac{4}{273}a^{6}+\frac{2}{39}a^{5}-\frac{44}{91}a^{4}+\frac{17}{273}a^{3}-\frac{17}{39}a^{2}-\frac{92}{273}a-\frac{8}{91}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

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:  $4$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{12}{91}a^{7}-\frac{53}{273}a^{6}+\frac{7}{39}a^{5}-\frac{20}{273}a^{4}-\frac{571}{273}a^{3}+\frac{43}{13}a^{2}+\frac{328}{273}a-\frac{379}{91}$, $\frac{2}{273}a^{7}-\frac{8}{273}a^{6}+\frac{4}{39}a^{5}+\frac{3}{91}a^{4}+\frac{34}{273}a^{3}+\frac{5}{39}a^{2}-\frac{184}{273}a+\frac{75}{91}$, $\frac{16}{273}a^{7}+\frac{9}{91}a^{6}+\frac{2}{13}a^{5}+\frac{163}{273}a^{4}-\frac{92}{273}a^{3}+\frac{1}{39}a^{2}+\frac{530}{273}a-\frac{128}{91}$, $\frac{22}{273}a^{7}+\frac{1}{91}a^{6}+\frac{5}{39}a^{5}+\frac{33}{91}a^{4}-\frac{118}{91}a^{3}+\frac{29}{39}a^{2}+\frac{524}{273}a-\frac{346}{273}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 81.706186763 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{3}\cdot 81.706186763 \cdot 1}{2\sqrt{8671400783}}\approx 0.43529122614$

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

\(\Q(\sqrt{17}) \), 4.2.13583.1

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

Sibling fields

Galois closure: 16.0.166101760110563345913601.2
Degree 8 sibling: 8.0.23973872753.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{2}$ ${\href{/padicField/3.2.0.1}{2} }^{4}$ ${\href{/padicField/5.8.0.1}{8} }$ ${\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }$ ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ R ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }$ ${\href{/padicField/29.8.0.1}{8} }$ ${\href{/padicField/31.8.0.1}{8} }$ ${\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }$ ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ R ${\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}$

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
\(17\) Copy content Toggle raw display 17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
\(47\) Copy content Toggle raw display $\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.17.2t1.a.a$1$ $ 17 $ \(\Q(\sqrt{17}) \) $C_2$ (as 2T1) $1$ $1$
1.47.2t1.a.a$1$ $ 47 $ \(\Q(\sqrt{-47}) \) $C_2$ (as 2T1) $1$ $-1$
1.799.2t1.a.a$1$ $ 17 \cdot 47 $ \(\Q(\sqrt{-799}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.799.4t3.a.a$2$ $ 17 \cdot 47 $ 4.2.13583.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.799.8t6.a.a$2$ $ 17 \cdot 47 $ 8.2.8671400783.1 $D_{8}$ (as 8T6) $1$ $0$
* 2.799.8t6.a.b$2$ $ 17 \cdot 47 $ 8.2.8671400783.1 $D_{8}$ (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 *.