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Properties

Label 16.0.81272760863...5561.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{6}\cdot 17^{14}$
Root discriminant $31.22$
Ramified primes $13, 17$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![52, -650, 2803, -3690, 5678, 4380, -7300, 776, 3245, -1026, -381, 198, 0, 16, -2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 2*x^14 + 16*x^13 + 198*x^11 - 381*x^10 - 1026*x^9 + 3245*x^8 + 776*x^7 - 7300*x^6 + 4380*x^5 + 5678*x^4 - 3690*x^3 + 2803*x^2 - 650*x + 52)
 
gp: K = bnfinit(x^16 - 4*x^15 - 2*x^14 + 16*x^13 + 198*x^11 - 381*x^10 - 1026*x^9 + 3245*x^8 + 776*x^7 - 7300*x^6 + 4380*x^5 + 5678*x^4 - 3690*x^3 + 2803*x^2 - 650*x + 52, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 2 x^{14} + 16 x^{13} + 198 x^{11} - 381 x^{10} - 1026 x^{9} + 3245 x^{8} + 776 x^{7} - 7300 x^{6} + 4380 x^{5} + 5678 x^{4} - 3690 x^{3} + 2803 x^{2} - 650 x + 52 \)

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:  \(812727608637355438705561=13^{6}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.22$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{32} a^{10} - \frac{3}{32} a^{8} - \frac{1}{8} a^{7} - \frac{1}{16} a^{5} + \frac{1}{32} a^{4} + \frac{1}{8} a^{3} + \frac{5}{32} a^{2} + \frac{1}{16} a - \frac{1}{8}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{3}{64} a^{9} - \frac{1}{64} a^{8} - \frac{1}{16} a^{7} - \frac{1}{32} a^{6} + \frac{11}{64} a^{5} + \frac{11}{64} a^{4} + \frac{1}{64} a^{3} - \frac{11}{64} a^{2} - \frac{3}{32} a + \frac{1}{16}$, $\frac{1}{128} a^{12} - \frac{1}{32} a^{9} - \frac{1}{128} a^{8} - \frac{3}{64} a^{7} - \frac{7}{128} a^{6} - \frac{9}{64} a^{5} + \frac{11}{64} a^{3} - \frac{45}{128} a^{2} - \frac{29}{64} a - \frac{3}{32}$, $\frac{1}{51712} a^{13} - \frac{21}{51712} a^{12} - \frac{9}{3232} a^{11} - \frac{137}{12928} a^{10} + \frac{2499}{51712} a^{9} + \frac{1967}{51712} a^{8} - \frac{2697}{51712} a^{7} - \frac{3679}{51712} a^{6} + \frac{3413}{25856} a^{5} + \frac{3483}{25856} a^{4} + \frac{13173}{51712} a^{3} - \frac{14537}{51712} a^{2} - \frac{1701}{25856} a - \frac{1217}{12928}$, $\frac{1}{206848} a^{14} - \frac{585}{206848} a^{12} + \frac{319}{51712} a^{11} - \frac{929}{206848} a^{10} - \frac{5905}{103424} a^{9} - \frac{8975}{103424} a^{8} - \frac{535}{51712} a^{7} - \frac{15489}{206848} a^{6} + \frac{4447}{25856} a^{5} + \frac{18867}{206848} a^{4} - \frac{71}{3232} a^{3} + \frac{12905}{206848} a^{2} - \frac{26843}{103424} a + \frac{18075}{51712}$, $\frac{1}{36777988096} a^{15} + \frac{39569}{36777988096} a^{14} + \frac{139623}{36777988096} a^{13} + \frac{84559795}{36777988096} a^{12} - \frac{91469669}{36777988096} a^{11} - \frac{120017683}{36777988096} a^{10} - \frac{47669661}{2298624256} a^{9} + \frac{1825909307}{18388994048} a^{8} + \frac{1140677427}{36777988096} a^{7} - \frac{1837477353}{36777988096} a^{6} - \frac{3322461557}{36777988096} a^{5} - \frac{4637033021}{36777988096} a^{4} - \frac{15015132775}{36777988096} a^{3} + \frac{786460161}{1935683584} a^{2} + \frac{6451783227}{18388994048} a + \frac{2379150235}{9194497024}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 188887.024999 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

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 $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.0.3757.1, 4.0.63869.1, 8.4.69347235737.1 x2, 8.0.4079249161.1

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 8 siblings: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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}$
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$