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Properties

Label 16.0.10749542400...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{10}$
Root discriminant $13.40$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^2.D_4$ (as 16T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -32, 116, -268, 476, -712, 890, -902, 757, -548, 344, -196, 101, -42, 16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 16*x^14 - 42*x^13 + 101*x^12 - 196*x^11 + 344*x^10 - 548*x^9 + 757*x^8 - 902*x^7 + 890*x^6 - 712*x^5 + 476*x^4 - 268*x^3 + 116*x^2 - 32*x + 4)
 
gp: K = bnfinit(x^16 - 4*x^15 + 16*x^14 - 42*x^13 + 101*x^12 - 196*x^11 + 344*x^10 - 548*x^9 + 757*x^8 - 902*x^7 + 890*x^6 - 712*x^5 + 476*x^4 - 268*x^3 + 116*x^2 - 32*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 16 x^{14} - 42 x^{13} + 101 x^{12} - 196 x^{11} + 344 x^{10} - 548 x^{9} + 757 x^{8} - 902 x^{7} + 890 x^{6} - 712 x^{5} + 476 x^{4} - 268 x^{3} + 116 x^{2} - 32 x + 4 \)

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:  \(1074954240000000000=2^{24}\cdot 3^{8}\cdot 5^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.40$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{10} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{42} a^{13} + \frac{1}{21} a^{12} + \frac{4}{21} a^{11} - \frac{1}{3} a^{10} + \frac{3}{14} a^{9} + \frac{2}{21} a^{8} + \frac{1}{7} a^{7} + \frac{4}{21} a^{6} + \frac{3}{14} a^{5} - \frac{10}{21} a^{4} - \frac{10}{21} a^{3} + \frac{1}{3} a - \frac{5}{21}$, $\frac{1}{420} a^{14} + \frac{8}{105} a^{12} - \frac{19}{70} a^{11} - \frac{11}{140} a^{10} + \frac{1}{15} a^{9} + \frac{52}{105} a^{8} + \frac{16}{35} a^{7} + \frac{3}{20} a^{6} - \frac{19}{210} a^{5} + \frac{73}{210} a^{4} - \frac{4}{105} a^{3} - \frac{13}{30} a^{2} - \frac{16}{35} a - \frac{17}{35}$, $\frac{1}{13020} a^{15} + \frac{1}{868} a^{14} + \frac{11}{6510} a^{13} + \frac{11}{2170} a^{12} - \frac{6023}{13020} a^{11} + \frac{5833}{13020} a^{10} - \frac{571}{6510} a^{9} + \frac{1553}{3255} a^{8} + \frac{923}{13020} a^{7} - \frac{4073}{13020} a^{6} + \frac{1633}{6510} a^{5} + \frac{2027}{6510} a^{4} - \frac{1301}{6510} a^{3} - \frac{2161}{6510} a^{2} - \frac{211}{3255} a - \frac{26}{217}$

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:  \( \frac{4853}{1860} a^{15} - \frac{13013}{1860} a^{14} + \frac{33786}{1085} a^{13} - \frac{420463}{6510} a^{12} + \frac{2099551}{13020} a^{11} - \frac{162329}{620} a^{10} + \frac{100096}{217} a^{9} - \frac{725653}{1085} a^{8} + \frac{10755361}{13020} a^{7} - \frac{11321171}{13020} a^{6} + \frac{2210666}{3255} a^{5} - \frac{2791331}{6510} a^{4} + \frac{1558427}{6510} a^{3} - \frac{5855}{62} a^{2} + \frac{1522}{155} a + \frac{14846}{3255} \) (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:  \( 1883.39472018 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2.D_4$ (as 16T92):

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 $C_2\times C_2^2.D_4$
Character table for $C_2\times C_2^2.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(i, \sqrt{15})\), 8.0.12960000.1, 8.0.115200000.2, 8.0.115200000.1

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
$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}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$