Properties

Label 4.0.256000.2
Degree $4$
Signature $[0, 2]$
Discriminant $2^{11}\cdot 5^{3}$
Root discriminant $22.49$
Ramified primes $2, 5$
Class number $20$
Class group $[20]$
Galois group $C_4$ (as 4T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![90, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^4 + 20*x^2 + 90)
 
gp: K = bnfinit(x^4 + 20*x^2 + 90, 1)
 

Normalized defining polynomial

\( x^{4} + 20 x^{2} + 90 \)

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

Invariants

Degree:  $4$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(256000=2^{11}\cdot 5^{3}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.49$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5$
magma: PrimeDivisors(Discriminant(K));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(80=2^{4}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{80}(1,·)$, $\chi_{80}(9,·)$, $\chi_{80}(53,·)$, $\chi_{80}(77,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$

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

Class group and class number

$C_{20}$, which has order $20$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $1$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental unit:  \( a^{2} + 13 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3.63689291846 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4$ (as 4T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 4
The 4 conjugacy class representatives for $C_4$
Character table for $C_4$

Intermediate fields

\(\Q(\sqrt{10}) \)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.4.11.10$x^{4} + 8 x + 6$$4$$1$$11$$C_4$$[3, 4]$
$5$5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e4_5.4t1.6c1$1$ $ 2^{4} \cdot 5 $ $x^{4} + 20 x^{2} + 90$ $C_4$ (as 4T1) $0$ $-1$
* 1.2e3_5.2t1.1c1$1$ $ 2^{3} \cdot 5 $ $x^{2} - 10$ $C_2$ (as 2T1) $1$ $1$
* 1.2e4_5.4t1.6c2$1$ $ 2^{4} \cdot 5 $ $x^{4} + 20 x^{2} + 90$ $C_4$ (as 4T1) $0$ $-1$

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 *.