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Properties

Label 16.16.5317838520...2816.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{32}\cdot 3^{12}\cdot 13^{12}$
Root discriminant $62.42$
Ramified primes $2, 3, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $Q_{16}$ (as 16T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21069, -217296, 770220, -880416, -609876, 1445736, 185526, -634260, 6103, 122968, -11158, -11264, 1666, 428, -82, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 82*x^14 + 428*x^13 + 1666*x^12 - 11264*x^11 - 11158*x^10 + 122968*x^9 + 6103*x^8 - 634260*x^7 + 185526*x^6 + 1445736*x^5 - 609876*x^4 - 880416*x^3 + 770220*x^2 - 217296*x + 21069)
 
gp: K = bnfinit(x^16 - 4*x^15 - 82*x^14 + 428*x^13 + 1666*x^12 - 11264*x^11 - 11158*x^10 + 122968*x^9 + 6103*x^8 - 634260*x^7 + 185526*x^6 + 1445736*x^5 - 609876*x^4 - 880416*x^3 + 770220*x^2 - 217296*x + 21069, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 82 x^{14} + 428 x^{13} + 1666 x^{12} - 11264 x^{11} - 11158 x^{10} + 122968 x^{9} + 6103 x^{8} - 634260 x^{7} + 185526 x^{6} + 1445736 x^{5} - 609876 x^{4} - 880416 x^{3} + 770220 x^{2} - 217296 x + 21069 \)

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:  \(53178385204239177923287842816=2^{32}\cdot 3^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.42$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13} a^{8} - \frac{2}{13} a^{7} - \frac{4}{13} a^{6} - \frac{2}{13} a^{5} + \frac{2}{13} a^{4} + \frac{6}{13} a^{3} + \frac{3}{13} a^{2} + \frac{2}{13} a + \frac{3}{13}$, $\frac{1}{13} a^{9} + \frac{5}{13} a^{7} + \frac{3}{13} a^{6} - \frac{2}{13} a^{5} - \frac{3}{13} a^{4} + \frac{2}{13} a^{3} - \frac{5}{13} a^{2} - \frac{6}{13} a + \frac{6}{13}$, $\frac{1}{13} a^{10} + \frac{5}{13} a^{6} - \frac{6}{13} a^{5} + \frac{5}{13} a^{4} + \frac{4}{13} a^{3} + \frac{5}{13} a^{2} - \frac{4}{13} a - \frac{2}{13}$, $\frac{1}{13} a^{11} + \frac{5}{13} a^{7} - \frac{6}{13} a^{6} + \frac{5}{13} a^{5} + \frac{4}{13} a^{4} + \frac{5}{13} a^{3} - \frac{4}{13} a^{2} - \frac{2}{13} a$, $\frac{1}{39} a^{12} - \frac{1}{39} a^{11} - \frac{1}{39} a^{10} - \frac{1}{39} a^{9} + \frac{1}{39} a^{8} - \frac{8}{39} a^{7} - \frac{7}{39} a^{6} - \frac{11}{39} a^{5} + \frac{4}{39} a^{4} + \frac{1}{13} a^{2} - \frac{3}{13} a - \frac{1}{13}$, $\frac{1}{39} a^{13} + \frac{1}{39} a^{11} + \frac{1}{39} a^{10} - \frac{1}{39} a^{8} - \frac{4}{13} a^{7} - \frac{2}{13} a^{6} + \frac{17}{39} a^{5} + \frac{4}{39} a^{4} - \frac{4}{13} a^{3} + \frac{5}{13} a^{2} - \frac{6}{13} a + \frac{3}{13}$, $\frac{1}{39} a^{14} - \frac{1}{39} a^{11} + \frac{1}{39} a^{10} - \frac{1}{39} a^{8} + \frac{2}{39} a^{7} - \frac{2}{13} a^{6} + \frac{5}{13} a^{5} - \frac{4}{39} a^{4} - \frac{2}{13} a^{3} - \frac{4}{13} a^{2} + \frac{3}{13} a$, $\frac{1}{2394500191678131254529123} a^{15} - \frac{249976637389082906527}{798166730559377084843041} a^{14} - \frac{28116999266087527049747}{2394500191678131254529123} a^{13} - \frac{26149279875525520011827}{2394500191678131254529123} a^{12} + \frac{6967335582174446137467}{798166730559377084843041} a^{11} + \frac{17050629720849940082}{14168640187444563636267} a^{10} - \frac{1179533176053939973016}{798166730559377084843041} a^{9} + \frac{7575613724057850834231}{798166730559377084843041} a^{8} - \frac{28753993398888424569979}{2394500191678131254529123} a^{7} - \frac{702722083674161069765564}{2394500191678131254529123} a^{6} - \frac{274913788725109613376553}{798166730559377084843041} a^{5} - \frac{314398800988398146185897}{798166730559377084843041} a^{4} - \frac{267203188037873543914294}{798166730559377084843041} a^{3} - \frac{379186859072385470846086}{798166730559377084843041} a^{2} + \frac{64897566238807244916893}{798166730559377084843041} a - \frac{220194726052327391849248}{798166730559377084843041}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 835254398.611 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_{16}$ (as 16T14):

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 $Q_{16}$
Character table for $Q_{16}$

Intermediate fields

\(\Q(\sqrt{39}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{3}, \sqrt{13})\), 4.4.8112.1 x2, 4.4.7488.1 x2, 8.8.9475854336.1

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

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$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}$