Properties

Label 4.2.8619.2
Degree $4$
Signature $[2, 1]$
Discriminant $-8619$
Root discriminant $9.64$
Ramified primes $3, 13, 17$
Class number $1$
Class group trivial
Galois group $D_{4}$ (as 4T3)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 2*x^3 + 2*x^2 - x - 3)
 
gp: K = bnfinit(x^4 - 2*x^3 + 2*x^2 - x - 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -1, 2, -2, 1]);
 

\(x^{4} - 2 x^{3} + 2 x^{2} - x - 3\)  Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $4$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 1]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-8619\)\(\medspace = -\,3\cdot 13^{2}\cdot 17\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $9.64$
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:  $3, 13, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$  Toggle raw display

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

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:  $2$
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:  \( a^{2} - a - 1 \),  \( a^{3} + 2 a + 2 \)  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 5.6894971291 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{1}\cdot 5.6894971291 \cdot 1}{2\sqrt{8619}}\approx 0.77011451021$

Galois group

$D_4$ (as 4T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 8
The 5 conjugacy class representatives for $D_{4}$
Character table for $D_{4}$

Intermediate fields

\(\Q(\sqrt{13}) \)

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

Sibling fields

Galois closure: 8.0.193220905761.3
Degree 4 sibling: 4.0.33813.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }$ R ${\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.2.0.1}{2} }^{2}$ R R ${\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }$ ${\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.13.2t1.a.a$1$ $ 13 $ \(\Q(\sqrt{13}) \) $C_2$ (as 2T1) $1$ $1$
1.51.2t1.a.a$1$ $ 3 \cdot 17 $ \(\Q(\sqrt{-51}) \) $C_2$ (as 2T1) $1$ $-1$
1.663.2t1.a.a$1$ $ 3 \cdot 13 \cdot 17 $ \(\Q(\sqrt{-663}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.663.4t3.b.a$2$ $ 3 \cdot 13 \cdot 17 $ 4.2.8619.2 $D_{4}$ (as 4T3) $1$ $0$

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 *.