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Properties

Label 16.8.60258267277...2144.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{50}\cdot 3^{8}\cdot 13^{8}$
Root discriminant $54.48$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times D_4):C_4$ (as 16T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28561, 0, -105456, 0, 22308, 0, 36504, 0, -4810, 0, -3744, 0, -84, 0, 24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^14 - 84*x^12 - 3744*x^10 - 4810*x^8 + 36504*x^6 + 22308*x^4 - 105456*x^2 + 28561)
 
gp: K = bnfinit(x^16 + 24*x^14 - 84*x^12 - 3744*x^10 - 4810*x^8 + 36504*x^6 + 22308*x^4 - 105456*x^2 + 28561, 1)
 

Normalized defining polynomial

\( x^{16} + 24 x^{14} - 84 x^{12} - 3744 x^{10} - 4810 x^{8} + 36504 x^{6} + 22308 x^{4} - 105456 x^{2} + 28561 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6025826727796418865889542144=2^{50}\cdot 3^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.48$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{26} a^{8} - \frac{1}{13} a^{6} - \frac{3}{13} a^{4} - \frac{1}{2}$, $\frac{1}{26} a^{9} - \frac{1}{13} a^{7} - \frac{3}{13} a^{5} - \frac{1}{2} a$, $\frac{1}{26} a^{10} + \frac{3}{26} a^{6} + \frac{1}{26} a^{4} - \frac{1}{2}$, $\frac{1}{52} a^{11} - \frac{1}{52} a^{10} - \frac{1}{52} a^{9} - \frac{1}{52} a^{8} - \frac{2}{13} a^{7} + \frac{3}{13} a^{6} + \frac{5}{13} a^{5} - \frac{2}{13} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{35828} a^{12} + \frac{181}{17914} a^{10} + \frac{163}{35828} a^{8} + \frac{227}{1378} a^{6} - \frac{1355}{2756} a^{4} - \frac{19}{53} a^{2} - \frac{21}{212}$, $\frac{1}{35828} a^{13} - \frac{327}{35828} a^{11} - \frac{1}{52} a^{10} - \frac{263}{17914} a^{9} - \frac{1}{52} a^{8} - \frac{72}{689} a^{7} + \frac{3}{13} a^{6} - \frac{401}{2756} a^{5} - \frac{2}{13} a^{4} + \frac{83}{212} a^{3} + \frac{1}{4} a^{2} - \frac{37}{106} a + \frac{1}{4}$, $\frac{1}{897871427596} a^{14} - \frac{9092813}{897871427596} a^{12} - \frac{7454982749}{897871427596} a^{10} - \frac{23971201}{1303151564} a^{8} - \frac{7389934571}{69067032892} a^{6} - \frac{2550688769}{5312848684} a^{4} - \frac{2281866377}{5312848684} a^{2} + \frac{31526015}{408680668}$, $\frac{1}{897871427596} a^{15} - \frac{9092813}{897871427596} a^{13} - \frac{7454982749}{897871427596} a^{11} - \frac{23971201}{1303151564} a^{9} - \frac{7389934571}{69067032892} a^{7} - \frac{2550688769}{5312848684} a^{5} - \frac{2281866377}{5312848684} a^{3} + \frac{31526015}{408680668} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 115725889.884 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4):C_4$ (as 16T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times D_4):C_4$
Character table for $(C_2\times D_4):C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.4.29952.1, 4.4.7488.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.3588489216.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$