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Properties

Label 16.0.18809679472...5696.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{4}\cdot 7^{12}$
Root discriminant $16.02$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $D_8:C_2$ (as 16T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23, -18, 21, 22, -64, 96, -83, 16, 75, -144, 151, -124, 82, -46, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 21*x^14 - 46*x^13 + 82*x^12 - 124*x^11 + 151*x^10 - 144*x^9 + 75*x^8 + 16*x^7 - 83*x^6 + 96*x^5 - 64*x^4 + 22*x^3 + 21*x^2 - 18*x + 23)
 
gp: K = bnfinit(x^16 - 6*x^15 + 21*x^14 - 46*x^13 + 82*x^12 - 124*x^11 + 151*x^10 - 144*x^9 + 75*x^8 + 16*x^7 - 83*x^6 + 96*x^5 - 64*x^4 + 22*x^3 + 21*x^2 - 18*x + 23, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 21 x^{14} - 46 x^{13} + 82 x^{12} - 124 x^{11} + 151 x^{10} - 144 x^{9} + 75 x^{8} + 16 x^{7} - 83 x^{6} + 96 x^{5} - 64 x^{4} + 22 x^{3} + 21 x^{2} - 18 x + 23 \)

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:  \(18809679472226205696=2^{24}\cdot 3^{4}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.02$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{127} a^{14} - \frac{58}{127} a^{13} - \frac{15}{127} a^{12} - \frac{50}{127} a^{11} - \frac{52}{127} a^{10} - \frac{14}{127} a^{9} - \frac{56}{127} a^{8} + \frac{30}{127} a^{7} + \frac{9}{127} a^{6} + \frac{63}{127} a^{5} + \frac{34}{127} a^{4} - \frac{19}{127} a^{3} + \frac{26}{127} a^{2} + \frac{16}{127} a - \frac{26}{127}$, $\frac{1}{328545670441} a^{15} - \frac{280420873}{328545670441} a^{14} - \frac{147509627401}{328545670441} a^{13} + \frac{100888253050}{328545670441} a^{12} - \frac{105595822684}{328545670441} a^{11} + \frac{132877875898}{328545670441} a^{10} - \frac{128660434405}{328545670441} a^{9} + \frac{2658969419}{7640596987} a^{8} + \frac{117232123253}{328545670441} a^{7} + \frac{3000586444}{7640596987} a^{6} + \frac{158911394994}{328545670441} a^{5} - \frac{137193369911}{328545670441} a^{4} - \frac{37070410675}{328545670441} a^{3} + \frac{60963262424}{328545670441} a^{2} - \frac{160829435747}{328545670441} a + \frac{2156511406}{14284594367}$

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:  \( 843.403644371 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8:C_2$ (as 16T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-14}) \), 4.2.448.1 x2, 4.0.392.1 x2, \(\Q(\sqrt{2}, \sqrt{-7})\), 8.0.9834496.2

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 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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$3$3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$