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Properties

Label 16.0.41137585140...3761.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{4}\cdot 13^{8}\cdot 53^{8}$
Root discriminant $34.55$
Ramified primes $3, 13, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![269, 496, 8110, 1625, -73, 6684, -953, -1528, 1583, -171, -25, 154, -72, 13, 11, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 11*x^14 + 13*x^13 - 72*x^12 + 154*x^11 - 25*x^10 - 171*x^9 + 1583*x^8 - 1528*x^7 - 953*x^6 + 6684*x^5 - 73*x^4 + 1625*x^3 + 8110*x^2 + 496*x + 269)
 
gp: K = bnfinit(x^16 - 5*x^15 + 11*x^14 + 13*x^13 - 72*x^12 + 154*x^11 - 25*x^10 - 171*x^9 + 1583*x^8 - 1528*x^7 - 953*x^6 + 6684*x^5 - 73*x^4 + 1625*x^3 + 8110*x^2 + 496*x + 269, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 11 x^{14} + 13 x^{13} - 72 x^{12} + 154 x^{11} - 25 x^{10} - 171 x^{9} + 1583 x^{8} - 1528 x^{7} - 953 x^{6} + 6684 x^{5} - 73 x^{4} + 1625 x^{3} + 8110 x^{2} + 496 x + 269 \)

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:  \(4113758514028199904823761=3^{4}\cdot 13^{8}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.55$
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:  $3, 13, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{99} a^{14} + \frac{4}{33} a^{13} + \frac{16}{99} a^{12} + \frac{1}{11} a^{11} - \frac{1}{99} a^{10} - \frac{4}{99} a^{9} + \frac{7}{99} a^{8} + \frac{2}{11} a^{7} - \frac{32}{99} a^{6} - \frac{1}{9} a^{5} + \frac{14}{99} a^{4} + \frac{4}{11} a^{3} - \frac{1}{33} a^{2} - \frac{13}{99} a + \frac{5}{99}$, $\frac{1}{3587848031617315349287041} a^{15} - \frac{3208575819521442942640}{3587848031617315349287041} a^{14} - \frac{561839661678964305074546}{3587848031617315349287041} a^{13} - \frac{261978282137129320733845}{3587848031617315349287041} a^{12} + \frac{508233109396293067044494}{3587848031617315349287041} a^{11} - \frac{55003554397316974347574}{398649781290812816587449} a^{10} + \frac{465592828743717215293442}{3587848031617315349287041} a^{9} - \frac{337101792699825928697068}{3587848031617315349287041} a^{8} + \frac{1627445841581904867862825}{3587848031617315349287041} a^{7} + \frac{180119282902541779355720}{398649781290812816587449} a^{6} - \frac{463459937684350217527826}{3587848031617315349287041} a^{5} + \frac{394461696410677424103679}{3587848031617315349287041} a^{4} - \frac{274689922252724485813673}{1195949343872438449762347} a^{3} - \frac{545063746383996665243950}{3587848031617315349287041} a^{2} - \frac{9162449848053163871306}{108722667624767131796577} a + \frac{1363798892022232767792874}{3587848031617315349287041}$

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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 370301.12164 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.8957.1, 8.0.38268683973.2, 8.0.4252075997.1, 8.8.2028240250569.1

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

Sibling fields

Degree 16 sibling: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$