# Properties

 Label 4.0.2433600.5 Degree $4$ Signature $[0, 2]$ Discriminant $2433600$ Root discriminant $$39.50$$ Ramified primes see page Class number $64$ Class group $[2, 4, 8]$ Galois group $C_2^2$ (as 4T2)

# Learn more

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 2*x^3 + x^2 + 390)

gp: K = bnfinit(x^4 - 2*x^3 + x^2 + 390, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![390, 0, 1, -2, 1]);

$$x^{4} - 2x^{3} + x^{2} + 390$$

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: $$2433600$$ 2433600 $$\medspace = 2^{6}\cdot 3^{2}\cdot 5^{2}\cdot 13^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$39.50$$ 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: $$2$$, $$3$$, $$5$$, $$13$$ 2, 3, 5, 13 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Gal(K/\Q) }$: $4$ This field is Galois and abelian over $\Q$. Conductor: $$1560=2^{3}\cdot 3\cdot 5\cdot 13$$ Dirichlet character group: $\lbrace$$\chi_{1560}(1,·), \chi_{1560}(1481,·), \chi_{1560}(469,·), \chi_{1560}(389,·)$$\rbrace$ This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{79}a^{3}+\frac{38}{79}a^{2}+\frac{20}{79}a+\frac{10}{79}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

 Monogenic: No Index: $1$ Inessential primes: None

## Class group and class number

$C_{2}\times C_{4}\times C_{8}$, which has order $64$

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: $$-1$$ -1  (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental unit: $\frac{2}{79}a^{3}-\frac{3}{79}a^{2}-\frac{39}{79}a+\frac{257}{79}$ 2/79*a^3 - 3/79*a^2 - 39/79*a + 257/79 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$3.63689291846$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{2}\cdot 3.63689291846 \cdot 64}{2\sqrt{2433600}}\approx 2.94520569061$

## Galois group

$C_2^2$ (as 4T2):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 An abelian group of order 4 The 4 conjugacy class representatives for $C_2^2$ Character table for $C_2^2$

## Intermediate fields

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

## Multiplicative Galois module structure

 $U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $A_1$

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R R R ${\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{2}$ R ${\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{2}$

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
$$2$$ 2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3] 2.2.3.2x^{2} + 6$$2$$1$$3$$C_2$$[3]$
$$3$$ 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2} 3.2.1.2x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$$5$$ 5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2} 5.2.1.2x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
$$13$$ 13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2} 13.2.1.1x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$