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Properties

Label 16.0.10136568828...3376.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 17^{4}\cdot 41^{4}$
Root discriminant $20.55$
Ramified primes $2, 17, 41$
Class number $2$
Class group $[2]$
Galois group $C_2\times D_4^2.C_2$ (as 16T602)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, 0, -130, 0, 281, 0, -430, 0, 396, 0, -198, 0, 57, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 + 57*x^12 - 198*x^10 + 396*x^8 - 430*x^6 + 281*x^4 - 130*x^2 + 49)
 
gp: K = bnfinit(x^16 - 10*x^14 + 57*x^12 - 198*x^10 + 396*x^8 - 430*x^6 + 281*x^4 - 130*x^2 + 49, 1)
 

Normalized defining polynomial

\( x^{16} - 10 x^{14} + 57 x^{12} - 198 x^{10} + 396 x^{8} - 430 x^{6} + 281 x^{4} - 130 x^{2} + 49 \)

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:  \(1013656882862280933376=2^{32}\cdot 17^{4}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.55$
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:  $2, 17, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{6} a^{8} + \frac{1}{6} a^{6} - \frac{1}{3} a^{4} - \frac{1}{6} a^{2} - \frac{1}{6}$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{7} - \frac{1}{3} a^{5} - \frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{3} a^{5} + \frac{1}{12} a^{4} - \frac{5}{12} a^{3} - \frac{5}{12} a + \frac{1}{12}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{1}{3} a^{4} - \frac{5}{12} a^{3} - \frac{5}{12} a^{2} + \frac{1}{6} a - \frac{5}{12}$, $\frac{1}{24} a^{12} - \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{1}{12} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{5}{12} a^{3} - \frac{1}{4} a^{2} - \frac{5}{12} a + \frac{5}{24}$, $\frac{1}{48} a^{13} - \frac{1}{48} a^{12} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{24} a^{9} + \frac{1}{24} a^{8} - \frac{1}{12} a^{7} - \frac{1}{6} a^{6} - \frac{11}{24} a^{5} + \frac{3}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{3}{16} a - \frac{19}{48}$, $\frac{1}{3504} a^{14} - \frac{7}{1168} a^{12} - \frac{1}{876} a^{10} - \frac{77}{1752} a^{8} - \frac{415}{1752} a^{6} + \frac{277}{876} a^{4} + \frac{941}{3504} a^{2} - \frac{1721}{3504}$, $\frac{1}{49056} a^{15} - \frac{1}{7008} a^{14} + \frac{139}{16352} a^{13} - \frac{125}{7008} a^{12} + \frac{3}{511} a^{11} - \frac{3}{73} a^{10} + \frac{23}{8176} a^{9} - \frac{23}{1168} a^{8} - \frac{5087}{24528} a^{7} + \frac{415}{3504} a^{6} + \frac{17}{1022} a^{5} + \frac{28}{73} a^{4} + \frac{21673}{49056} a^{3} + \frac{173}{2336} a^{2} + \frac{3097}{49056} a - \frac{2221}{7008}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 8657.31768557 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4^2.C_2$ (as 16T602):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2\times D_4^2.C_2$
Character table for $C_2\times D_4^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.1088.2, 4.4.2624.1, 4.0.44608.5, 8.4.1872822272.1, 8.4.1872822272.2, 8.0.1989873664.4

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$