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Properties

Label 8.2.107171875.1
Degree $8$
Signature $[2, 3]$
Discriminant $-107171875$
Root discriminant \(10.09\)
Ramified primes see page
Class number $1$
Class group trivial
Galois group $QD_{16}$ (as 8T8)

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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 + 2*x^6 + x^4 + 5*x^3 - 7*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^8 - 2*x^7 + 2*x^6 + x^4 + 5*x^3 - 7*x^2 - 6*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -7, 5, 1, 0, 2, -2, 1]);
 

\( x^{8} - 2x^{7} + 2x^{6} + x^{4} + 5x^{3} - 7x^{2} - 6x + 1 \) 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:   \(-107171875\) \(\medspace = -\,5^{6}\cdot 19^{3}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(10.09\)
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:   \(5\), \(19\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{131}a^{7}-\frac{60}{131}a^{6}-\frac{55}{131}a^{5}+\frac{46}{131}a^{4}-\frac{47}{131}a^{3}-\frac{20}{131}a^{2}-\frac{26}{131}a+\frac{61}{131}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

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:   $\frac{80}{131}a^{7}-\frac{215}{131}a^{6}+\frac{316}{131}a^{5}-\frac{250}{131}a^{4}+\frac{301}{131}a^{3}+\frac{103}{131}a^{2}-\frac{639}{131}a-\frac{98}{131}$, $\frac{25}{131}a^{7}-\frac{59}{131}a^{6}+\frac{66}{131}a^{5}-\frac{29}{131}a^{4}+\frac{4}{131}a^{3}+\frac{155}{131}a^{2}-\frac{257}{131}a-\frac{47}{131}$, $\frac{28}{131}a^{7}-\frac{108}{131}a^{6}+\frac{163}{131}a^{5}-\frac{153}{131}a^{4}+\frac{125}{131}a^{3}-\frac{36}{131}a^{2}-\frac{335}{131}a+\frac{5}{131}$, $\frac{50}{131}a^{7}-\frac{118}{131}a^{6}+\frac{132}{131}a^{5}-\frac{58}{131}a^{4}+\frac{139}{131}a^{3}+\frac{179}{131}a^{2}-\frac{383}{131}a-\frac{225}{131}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 6.79191421448 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{3}\cdot 6.79191421448 \cdot 1}{2\sqrt{107171875}}\approx 0.325477804789$

Galois group

$\SD_{16}$ (as 8T8):

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 $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.475.1

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

Sibling fields

Galois closure: 16.0.4146377695556640625.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }$ ${\href{/padicField/3.8.0.1}{8} }$ R ${\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }$ ${\href{/padicField/17.4.0.1}{4} }^{2}$ R ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }$ ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }$ ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(5\) Copy content Toggle raw display 5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
\(19\) Copy content Toggle raw display $\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.19.2t1.a.a$1$ $ 19 $ \(\Q(\sqrt{-19}) \) $C_2$ (as 2T1) $1$ $-1$
1.95.2t1.a.a$1$ $ 5 \cdot 19 $ \(\Q(\sqrt{-95}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.95.4t3.c.a$2$ $ 5 \cdot 19 $ 4.2.475.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.475.8t8.a.a$2$ $ 5^{2} \cdot 19 $ 8.2.107171875.1 $QD_{16}$ (as 8T8) $0$ $0$
* 2.475.8t8.a.b$2$ $ 5^{2} \cdot 19 $ 8.2.107171875.1 $QD_{16}$ (as 8T8) $0$ $0$

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 *.