Properties

Label 4.0.247009.2
Degree $4$
Signature $[0, 2]$
Discriminant $247009$
Root discriminant $22.29$
Ramified primes $7, 71$
Class number $7$
Class group $[7]$
Galois group $C_2^2$ (as 4T2)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^4 + 39*x^2 + 256)
 
gp: K = bnfinit(x^4 + 39*x^2 + 256, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, 0, 39, 0, 1]);
 

\(x^{4} + 39 x^{2} + 256\)  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:  $[0, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(247009\)\(\medspace = 7^{2}\cdot 71^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $22.29$
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:  $7, 71$
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:  \(497=7\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{497}(496,·)$, $\chi_{497}(1,·)$, $\chi_{497}(356,·)$, $\chi_{497}(141,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{3} - \frac{9}{32} a - \frac{1}{2}$  Toggle raw display

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

Class group and class number

$C_{7}$, which has order $7$

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 $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental unit:  \( \frac{201}{16} a^{3} + \frac{1679}{16} a \)  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 14.6925505602 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{2}\cdot 14.6925505602 \cdot 7}{2\sqrt{247009}}\approx 4.08477920203$

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

\(\Q(\sqrt{497}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-71}) \)

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 ${\href{/padicField/2.1.0.1}{1} }^{4}$ ${\href{/padicField/3.2.0.1}{2} }^{2}$ ${\href{/padicField/5.2.0.1}{2} }^{2}$ R ${\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{2}$

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
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$71$71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$