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Properties

Label 16.16.1450531566...6641.1
Degree $16$
Signature $[16, 0]$
Discriminant $13^{12}\cdot 53^{8}$
Root discriminant $49.84$
Ramified primes $13, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -24, 164, -56, -2683, 7321, -4418, -5231, 5943, 1058, -2481, -41, 489, 4, -44, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 44*x^14 + 4*x^13 + 489*x^12 - 41*x^11 - 2481*x^10 + 1058*x^9 + 5943*x^8 - 5231*x^7 - 4418*x^6 + 7321*x^5 - 2683*x^4 - 56*x^3 + 164*x^2 - 24*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 44*x^14 + 4*x^13 + 489*x^12 - 41*x^11 - 2481*x^10 + 1058*x^9 + 5943*x^8 - 5231*x^7 - 4418*x^6 + 7321*x^5 - 2683*x^4 - 56*x^3 + 164*x^2 - 24*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 44 x^{14} + 4 x^{13} + 489 x^{12} - 41 x^{11} - 2481 x^{10} + 1058 x^{9} + 5943 x^{8} - 5231 x^{7} - 4418 x^{6} + 7321 x^{5} - 2683 x^{4} - 56 x^{3} + 164 x^{2} - 24 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1450531566903202684958906641=13^{12}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.84$
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:  $13, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{4}{9} a^{6} - \frac{2}{9} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{1}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{4}{9} a^{6} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{81} a^{14} + \frac{2}{81} a^{12} - \frac{1}{81} a^{11} + \frac{4}{27} a^{10} + \frac{1}{9} a^{9} - \frac{4}{27} a^{8} - \frac{1}{81} a^{7} + \frac{7}{81} a^{6} - \frac{34}{81} a^{5} - \frac{5}{27} a^{4} + \frac{10}{27} a^{3} - \frac{28}{81} a^{2} - \frac{28}{81} a + \frac{17}{81}$, $\frac{1}{305289} a^{15} - \frac{113}{101763} a^{14} + \frac{4898}{305289} a^{13} + \frac{11507}{305289} a^{12} - \frac{2828}{33921} a^{11} - \frac{3457}{33921} a^{10} - \frac{7201}{101763} a^{9} - \frac{45667}{305289} a^{8} - \frac{49640}{305289} a^{7} - \frac{127750}{305289} a^{6} + \frac{9389}{33921} a^{5} - \frac{8291}{101763} a^{4} - \frac{21583}{305289} a^{3} - \frac{76135}{305289} a^{2} - \frac{13000}{305289} a + \frac{26849}{101763}$

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:  $15$
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:  \( 516194249.44 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{689}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{13}, \sqrt{53})\), 4.4.8957.1 x2, 4.4.36517.1 x2, 8.8.225360027841.1, 8.8.718600843493.1 x4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$