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Properties

Label 16.0.33672290266...6576.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 23^{8}$
Root discriminant $16.61$
Ramified primes $2, 3, 23$
Class number $3$
Class group $[3]$
Galois group $D_4\times C_2$ (as 16T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, 0, 192, 0, -112, 0, 96, 0, 204, 0, 24, 0, -7, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 3*x^14 - 7*x^12 + 24*x^10 + 204*x^8 + 96*x^6 - 112*x^4 + 192*x^2 + 256)
 
gp: K = bnfinit(x^16 + 3*x^14 - 7*x^12 + 24*x^10 + 204*x^8 + 96*x^6 - 112*x^4 + 192*x^2 + 256, 1)
 

Normalized defining polynomial

\( x^{16} + 3 x^{14} - 7 x^{12} + 24 x^{10} + 204 x^{8} + 96 x^{6} - 112 x^{4} + 192 x^{2} + 256 \)

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:  \(33672290266555416576=2^{16}\cdot 3^{8}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.61$
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 over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{8} - \frac{1}{4} a^{6} - \frac{5}{12} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{24} a^{9} - \frac{1}{8} a^{7} + \frac{7}{24} a^{5} - \frac{1}{4} a^{3} + \frac{1}{6} a$, $\frac{1}{48} a^{10} + \frac{1}{48} a^{8} - \frac{5}{48} a^{6} - \frac{1}{2} a^{5} + \frac{11}{24} a^{4} - \frac{1}{2} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{96} a^{11} + \frac{1}{96} a^{9} - \frac{5}{96} a^{7} - \frac{1}{4} a^{6} - \frac{13}{48} a^{5} + \frac{1}{4} a^{4} - \frac{5}{24} a^{3} - \frac{1}{4} a^{2} - \frac{1}{3} a$, $\frac{1}{3168} a^{12} - \frac{1}{96} a^{10} + \frac{1}{96} a^{8} - \frac{1}{4} a^{7} + \frac{23}{132} a^{6} - \frac{1}{4} a^{5} + \frac{5}{12} a^{4} + \frac{1}{4} a^{3} + \frac{1}{12} a^{2} - \frac{1}{2} a + \frac{46}{99}$, $\frac{1}{6336} a^{13} - \frac{1}{192} a^{11} + \frac{1}{192} a^{9} - \frac{1}{24} a^{8} - \frac{43}{264} a^{7} + \frac{1}{8} a^{6} + \frac{11}{24} a^{5} + \frac{5}{24} a^{4} - \frac{5}{24} a^{3} - \frac{1}{4} a^{2} - \frac{53}{198} a + \frac{1}{3}$, $\frac{1}{6336} a^{14} - \frac{1}{6336} a^{12} + \frac{1}{192} a^{10} + \frac{1}{264} a^{8} - \frac{23}{264} a^{6} - \frac{1}{2} a^{5} - \frac{1}{24} a^{4} - \frac{1}{2} a^{3} - \frac{53}{198} a^{2} - \frac{1}{2} a + \frac{43}{99}$, $\frac{1}{12672} a^{15} - \frac{1}{12672} a^{13} + \frac{1}{384} a^{11} + \frac{1}{528} a^{9} - \frac{23}{528} a^{7} - \frac{1}{4} a^{6} + \frac{23}{48} a^{5} + \frac{1}{4} a^{4} - \frac{53}{396} a^{3} - \frac{1}{4} a^{2} + \frac{43}{198} a$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( \frac{113}{12672} a^{15} + \frac{27}{1408} a^{13} - \frac{31}{384} a^{11} + \frac{73}{264} a^{9} + \frac{1663}{1056} a^{7} - \frac{13}{24} a^{5} - \frac{593}{792} a^{3} + \frac{103}{66} a \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6753.06260878 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4$ (as 16T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{69}) \), \(\Q(\sqrt{-69}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{-23}) \), \(\Q(i, \sqrt{69})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{23})\), \(\Q(\sqrt{3}, \sqrt{23})\), \(\Q(\sqrt{-3}, \sqrt{-23})\), \(\Q(\sqrt{3}, \sqrt{-23})\), \(\Q(\sqrt{-3}, \sqrt{23})\), 4.2.3312.1 x2, 4.2.3312.2 x2, 4.0.6348.1 x2, 4.0.6348.2 x2, 8.0.5802782976.1, 8.0.10969344.1 x2, 8.0.644753664.3 x2, 8.4.5802782976.1 x2, 8.0.362673936.1 x2, 8.0.5802782976.3, 8.0.5802782976.2

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

Sibling fields

Degree 8 siblings: data not computed

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{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}$