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Properties

Label 16.0.28179280429...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{8}\cdot 5^{8}$
Root discriminant $21.91$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[4]$
Galois group $Q_8 : C_2^2$ (as 16T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 32, 232, -496, 512, -136, -238, 320, -128, -64, 120, -80, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 120*x^12 - 64*x^11 - 128*x^10 + 320*x^9 - 238*x^8 - 136*x^7 + 512*x^6 - 496*x^5 + 232*x^4 + 32*x^3 + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 120*x^12 - 64*x^11 - 128*x^10 + 320*x^9 - 238*x^8 - 136*x^7 + 512*x^6 - 496*x^5 + 232*x^4 + 32*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 80 x^{13} + 120 x^{12} - 64 x^{11} - 128 x^{10} + 320 x^{9} - 238 x^{8} - 136 x^{7} + 512 x^{6} - 496 x^{5} + 232 x^{4} + 32 x^{3} + 1 \)

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:  \(2817928042905600000000=2^{40}\cdot 3^{8}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.91$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{8} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{8} a^{13} - \frac{1}{2} a^{7} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{280} a^{14} + \frac{1}{20} a^{13} + \frac{1}{20} a^{12} + \frac{1}{70} a^{11} - \frac{2}{35} a^{10} - \frac{1}{56} a^{9} + \frac{1}{35} a^{8} + \frac{12}{35} a^{7} + \frac{1}{280} a^{6} - \frac{5}{28} a^{5} + \frac{33}{140} a^{4} - \frac{13}{35} a^{3} - \frac{13}{70} a^{2} - \frac{73}{280} a + \frac{2}{35}$, $\frac{1}{32084080} a^{15} - \frac{7039}{6416816} a^{14} - \frac{267481}{4583440} a^{13} - \frac{76527}{32084080} a^{12} - \frac{702847}{32084080} a^{11} - \frac{264633}{4583440} a^{10} - \frac{37879}{1394960} a^{9} + \frac{320319}{32084080} a^{8} - \frac{9781983}{32084080} a^{7} - \frac{303517}{4583440} a^{6} - \frac{1978919}{32084080} a^{5} + \frac{13206337}{32084080} a^{4} - \frac{5503031}{32084080} a^{3} + \frac{2141653}{6416816} a^{2} + \frac{5204583}{32084080} a - \frac{6671089}{32084080}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( \frac{127761}{2291720} a^{15} - \frac{881271}{2291720} a^{14} + \frac{2961849}{2291720} a^{13} - \frac{5730311}{2291720} a^{12} + \frac{4154289}{2291720} a^{11} + \frac{1697533}{458344} a^{10} - \frac{1089979}{99640} a^{9} + \frac{23032231}{2291720} a^{8} + \frac{13404801}{2291720} a^{7} - \frac{9897259}{458344} a^{6} + \frac{46589601}{2291720} a^{5} + \frac{6222161}{2291720} a^{4} - \frac{39049127}{2291720} a^{3} + \frac{37140217}{2291720} a^{2} + \frac{2233211}{2291720} a - \frac{25013}{458344} \) (order $4$)
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:  \( 37781.5309275 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2$ (as 16T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 17 conjugacy class representatives for $Q_8 : C_2^2$
Character table for $Q_8 : C_2^2$

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(i, \sqrt{15})\), 8.0.3317760000.5, 8.0.2123366400.7 x2, 8.0.53084160000.1 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 R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.20.59$x^{8} + 4 x^{5} + 2 x^{4} + 10$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
2.8.20.59$x^{8} + 4 x^{5} + 2 x^{4} + 10$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
$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}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$