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Properties

Label 16.0.97706917712...3153.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{7}\cdot 47^{8}$
Root discriminant $23.68$
Ramified primes $17, 47$
Class number $5$
Class group $[5]$
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51, -68, 162, -21, 36, 90, -42, 80, -48, -36, 64, -24, -18, 19, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 19*x^13 - 18*x^12 - 24*x^11 + 64*x^10 - 36*x^9 - 48*x^8 + 80*x^7 - 42*x^6 + 90*x^5 + 36*x^4 - 21*x^3 + 162*x^2 - 68*x + 51)
 
gp: K = bnfinit(x^16 - 3*x^15 + 19*x^13 - 18*x^12 - 24*x^11 + 64*x^10 - 36*x^9 - 48*x^8 + 80*x^7 - 42*x^6 + 90*x^5 + 36*x^4 - 21*x^3 + 162*x^2 - 68*x + 51, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 19 x^{13} - 18 x^{12} - 24 x^{11} + 64 x^{10} - 36 x^{9} - 48 x^{8} + 80 x^{7} - 42 x^{6} + 90 x^{5} + 36 x^{4} - 21 x^{3} + 162 x^{2} - 68 x + 51 \)

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:  \(9770691771209608583153=17^{7}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.68$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{7} - \frac{1}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a^{2} + \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{351} a^{13} + \frac{1}{27} a^{12} + \frac{1}{117} a^{11} - \frac{38}{351} a^{10} - \frac{58}{351} a^{9} - \frac{29}{351} a^{8} + \frac{49}{351} a^{7} + \frac{17}{351} a^{6} - \frac{16}{351} a^{5} + \frac{20}{117} a^{4} - \frac{5}{13} a^{3} + \frac{167}{351} a^{2} + \frac{98}{351} a - \frac{35}{117}$, $\frac{1}{4563} a^{14} + \frac{1}{4563} a^{13} - \frac{77}{1521} a^{12} + \frac{160}{4563} a^{11} - \frac{694}{4563} a^{10} - \frac{503}{4563} a^{9} + \frac{709}{4563} a^{8} - \frac{25}{4563} a^{7} + \frac{365}{4563} a^{6} + \frac{383}{1521} a^{5} + \frac{22}{507} a^{4} + \frac{1592}{4563} a^{3} + \frac{1877}{4563} a^{2} - \frac{242}{507} a + \frac{244}{507}$, $\frac{1}{1099683} a^{15} - \frac{103}{1099683} a^{14} + \frac{419}{1099683} a^{13} + \frac{20297}{1099683} a^{12} - \frac{3590}{122187} a^{11} - \frac{88799}{1099683} a^{10} + \frac{47314}{1099683} a^{9} + \frac{91456}{1099683} a^{8} - \frac{157312}{1099683} a^{7} - \frac{40724}{1099683} a^{6} - \frac{508570}{1099683} a^{5} + \frac{350720}{1099683} a^{4} + \frac{407815}{1099683} a^{3} + \frac{229690}{1099683} a^{2} + \frac{87632}{1099683} a - \frac{5624}{28197}$

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

Class group and class number

$C_{5}$, which has order $5$

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:  \( 17824.0536167 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{-47}) \), 4.0.37553.1, 8.0.23973872753.1

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 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ $16$ $16$ $16$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$