# Properties

 Label 4.0.18000.1 Degree $4$ Signature $[0, 2]$ Discriminant $2^{4}\cdot 3^{2}\cdot 5^{3}$ Root discriminant $11.58$ Ramified primes $2, 3, 5$ Class number $4$ Class group $[2, 2]$ Galois group $C_4$ (as 4T1)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^4 + 15*x^2 + 45)

gp: K = bnfinit(x^4 + 15*x^2 + 45, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45, 0, 15, 0, 1]);

## Normalizeddefining polynomial

$$x^{4} + 15 x^{2} + 45$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $4$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 2]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$18000=2^{4}\cdot 3^{2}\cdot 5^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $11.58$ 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: $2, 3, 5$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $4$ This field is Galois and abelian over $\Q$. Conductor: $$60=2^{2}\cdot 3\cdot 5$$ Dirichlet character group: $\lbrace$$\chi_{60}(1,·), \chi_{60}(47,·), \chi_{60}(49,·), \chi_{60}(23,·)$$\rbrace$ This is a CM field.

## Integral basis (with respect to field generator $$a$$)

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}\times C_{2}$, which has order $4$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $1$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental unit: $$\frac{1}{3} a^{2} + 2$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$0.962423650119$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$C_4$ (as 4T1):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A cyclic group of order 4 The 4 conjugacy class representatives for $C_4$ Character table for $C_4$

## Intermediate fields

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

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2} 33.4.2.2x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.60.4t1.a.a$1$ $2^{2} \cdot 3 \cdot 5$ $x^{4} + 15 x^{2} + 45$ $C_4$ (as 4T1) $0$ $-1$
* 1.5.2t1.a.a$1$ $5$ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.60.4t1.a.b$1$ $2^{2} \cdot 3 \cdot 5$ $x^{4} + 15 x^{2} + 45$ $C_4$ (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 *.