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Properties

Label 16.0.22067472149...4736.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 23^{8}$
Root discriminant $33.23$
Ramified primes $2, 3, 23$
Class number $48$ (GRH)
Class group $[4, 12]$ (GRH)
Galois group $C_2^4$ (as 16T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1679616, 0, 0, 0, -63504, 0, 0, 0, 1105, 0, 0, 0, -49, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 49*x^12 + 1105*x^8 - 63504*x^4 + 1679616)
 
gp: K = bnfinit(x^16 - 49*x^12 + 1105*x^8 - 63504*x^4 + 1679616, 1)
 

Normalized defining polynomial

\( x^{16} - 49 x^{12} + 1105 x^{8} - 63504 x^{4} + 1679616 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2206747214908975780724736=2^{32}\cdot 3^{8}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.23$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(552=2^{3}\cdot 3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{552}(1,·)$, $\chi_{552}(323,·)$, $\chi_{552}(137,·)$, $\chi_{552}(139,·)$, $\chi_{552}(461,·)$, $\chi_{552}(275,·)$, $\chi_{552}(277,·)$, $\chi_{552}(185,·)$, $\chi_{552}(47,·)$, $\chi_{552}(413,·)$, $\chi_{552}(415,·)$, $\chi_{552}(91,·)$, $\chi_{552}(229,·)$, $\chi_{552}(551,·)$, $\chi_{552}(367,·)$, $\chi_{552}(505,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11} a^{8} + \frac{3}{11} a^{4} - \frac{2}{11}$, $\frac{1}{66} a^{9} - \frac{19}{66} a^{5} + \frac{31}{66} a$, $\frac{1}{396} a^{10} - \frac{85}{396} a^{6} + \frac{97}{396} a^{2}$, $\frac{1}{2376} a^{11} - \frac{481}{2376} a^{7} + \frac{889}{2376} a^{3}$, $\frac{1}{15752880} a^{12} - \frac{529}{14256} a^{8} + \frac{2593}{14256} a^{4} - \frac{5574}{12155}$, $\frac{1}{94517280} a^{13} - \frac{529}{85536} a^{9} + \frac{16849}{85536} a^{5} + \frac{9368}{36465} a$, $\frac{1}{567103680} a^{14} - \frac{529}{513216} a^{10} + \frac{102385}{513216} a^{6} + \frac{4684}{109395} a^{2}$, $\frac{1}{3402622080} a^{15} - \frac{529}{3079296} a^{11} - \frac{410831}{3079296} a^{7} + \frac{114079}{656370} a^{3}$

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

Class group and class number

$C_{4}\times C_{12}$, which has order $48$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{5059}{3402622080} a^{15} + \frac{29}{3079296} a^{11} + \frac{5059}{3079296} a^{7} - \frac{247891}{2625480} a^{3} \) (order $24$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 207978.296439 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4$ (as 16T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_2^4$
Character table for $C_2^4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{138}) \), \(\Q(\sqrt{-138}) \), \(\Q(\sqrt{69}) \), \(\Q(\sqrt{-69}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{46}) \), \(\Q(\sqrt{-46}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(i, \sqrt{138})\), \(\Q(i, \sqrt{69})\), \(\Q(i, \sqrt{23})\), \(\Q(i, \sqrt{46})\), \(\Q(\sqrt{2}, \sqrt{69})\), \(\Q(\sqrt{2}, \sqrt{-69})\), \(\Q(\sqrt{2}, \sqrt{23})\), \(\Q(\sqrt{2}, \sqrt{-23})\), \(\Q(\sqrt{-2}, \sqrt{-69})\), \(\Q(\sqrt{-2}, \sqrt{69})\), \(\Q(\sqrt{-2}, \sqrt{23})\), \(\Q(\sqrt{-2}, \sqrt{-23})\), \(\Q(\sqrt{6}, \sqrt{23})\), \(\Q(\sqrt{6}, \sqrt{-23})\), \(\Q(\sqrt{6}, \sqrt{46})\), \(\Q(\sqrt{6}, \sqrt{-46})\), \(\Q(\sqrt{-6}, \sqrt{-23})\), \(\Q(\sqrt{-6}, \sqrt{23})\), \(\Q(\sqrt{-6}, \sqrt{-46})\), \(\Q(\sqrt{-6}, \sqrt{46})\), \(\Q(\sqrt{3}, \sqrt{46})\), \(\Q(\sqrt{3}, \sqrt{-46})\), \(\Q(\sqrt{3}, \sqrt{23})\), \(\Q(\sqrt{3}, \sqrt{-23})\), \(\Q(\sqrt{-3}, \sqrt{-46})\), \(\Q(\sqrt{-3}, \sqrt{46})\), \(\Q(\sqrt{-3}, \sqrt{-23})\), \(\Q(\sqrt{-3}, \sqrt{23})\), \(\Q(\zeta_{24})\), 8.0.1485512441856.8, 8.0.18339659776.1, 8.0.1485512441856.3, 8.0.1485512441856.2, 8.0.1485512441856.9, 8.0.5802782976.1, 8.8.1485512441856.2, 8.0.1485512441856.5, 8.0.92844527616.2, 8.0.1485512441856.7, 8.0.1485512441856.6, 8.0.92844527616.1, 8.0.1485512441856.1, 8.0.1485512441856.4

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 R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$