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Properties

Label 16.0.11984260029...8144.8
Degree $16$
Signature $[0, 8]$
Discriminant $2^{34}\cdot 17^{8}$
Root discriminant $17.99$
Ramified primes $2, 17$
Class number $1$
Class group Trivial
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, 0, 92, -104, 384, -544, 723, -598, 383, -112, -24, 56, -40, 8, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + x^14 + 8*x^13 - 40*x^12 + 56*x^11 - 24*x^10 - 112*x^9 + 383*x^8 - 598*x^7 + 723*x^6 - 544*x^5 + 384*x^4 - 104*x^3 + 92*x^2 + 32)
 
gp: K = bnfinit(x^16 - 2*x^15 + x^14 + 8*x^13 - 40*x^12 + 56*x^11 - 24*x^10 - 112*x^9 + 383*x^8 - 598*x^7 + 723*x^6 - 544*x^5 + 384*x^4 - 104*x^3 + 92*x^2 + 32, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + x^{14} + 8 x^{13} - 40 x^{12} + 56 x^{11} - 24 x^{10} - 112 x^{9} + 383 x^{8} - 598 x^{7} + 723 x^{6} - 544 x^{5} + 384 x^{4} - 104 x^{3} + 92 x^{2} + 32 \)

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:  \(119842600295694598144=2^{34}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.99$
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:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{10} + \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{3}{16} a^{7} - \frac{1}{16} a^{6} - \frac{3}{16} a^{5} - \frac{3}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{8} a^{2}$, $\frac{1}{48} a^{12} - \frac{1}{12} a^{10} + \frac{1}{12} a^{9} - \frac{1}{24} a^{8} - \frac{1}{6} a^{7} - \frac{1}{8} a^{6} - \frac{1}{12} a^{5} + \frac{17}{48} a^{4} - \frac{1}{2} a^{3} - \frac{7}{24} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{96} a^{13} - \frac{1}{96} a^{12} + \frac{1}{48} a^{11} + \frac{1}{48} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{12} a^{7} + \frac{1}{12} a^{6} - \frac{11}{32} a^{5} + \frac{25}{96} a^{4} + \frac{5}{48} a^{3} + \frac{17}{48} a^{2} - \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{96} a^{14} - \frac{1}{96} a^{12} - \frac{1}{48} a^{11} - \frac{1}{12} a^{10} + \frac{5}{48} a^{9} - \frac{1}{16} a^{8} + \frac{7}{48} a^{7} - \frac{7}{96} a^{6} - \frac{5}{16} a^{5} + \frac{19}{96} a^{4} - \frac{1}{6} a^{3} - \frac{19}{48} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{299263584} a^{15} + \frac{276121}{149631792} a^{14} + \frac{458825}{149631792} a^{13} - \frac{1882277}{299263584} a^{12} + \frac{106565}{9351987} a^{11} + \frac{4000889}{49877264} a^{10} + \frac{244015}{3117329} a^{9} + \frac{725697}{12469316} a^{8} + \frac{26764151}{299263584} a^{7} + \frac{92377}{149631792} a^{6} + \frac{8684187}{49877264} a^{5} - \frac{97895479}{299263584} a^{4} - \frac{24940433}{74815896} a^{3} - \frac{28382999}{149631792} a^{2} - \frac{1412115}{3117329} a - \frac{2963191}{18703974}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -1 \) (order $2$)
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:  \( 76945.8851596 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), 4.0.2312.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 4.0.1088.2 x2, 8.0.1368408064.3 x2, 8.0.342102016.2, 8.4.10947264512.1 x2

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$