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Properties

Label 16.0.81272760863...5561.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{6}\cdot 17^{14}$
Root discriminant $31.22$
Ramified primes $13, 17$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -72, 344, 1060, 3189, 3737, 3818, 1282, 557, -152, 335, 33, 45, -35, 25, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 25*x^14 - 35*x^13 + 45*x^12 + 33*x^11 + 335*x^10 - 152*x^9 + 557*x^8 + 1282*x^7 + 3818*x^6 + 3737*x^5 + 3189*x^4 + 1060*x^3 + 344*x^2 - 72*x + 16)
 
gp: K = bnfinit(x^16 - 6*x^15 + 25*x^14 - 35*x^13 + 45*x^12 + 33*x^11 + 335*x^10 - 152*x^9 + 557*x^8 + 1282*x^7 + 3818*x^6 + 3737*x^5 + 3189*x^4 + 1060*x^3 + 344*x^2 - 72*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 25 x^{14} - 35 x^{13} + 45 x^{12} + 33 x^{11} + 335 x^{10} - 152 x^{9} + 557 x^{8} + 1282 x^{7} + 3818 x^{6} + 3737 x^{5} + 3189 x^{4} + 1060 x^{3} + 344 x^{2} - 72 x + 16 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2588} a^{14} - \frac{60}{647} a^{13} + \frac{101}{2588} a^{12} - \frac{405}{2588} a^{11} - \frac{293}{2588} a^{10} + \frac{451}{2588} a^{9} - \frac{279}{2588} a^{8} - \frac{463}{1294} a^{7} + \frac{237}{2588} a^{6} + \frac{140}{647} a^{5} - \frac{175}{1294} a^{4} + \frac{1161}{2588} a^{3} + \frac{87}{2588} a^{2} - \frac{313}{1294} a + \frac{243}{647}$, $\frac{1}{11049481174548718856} a^{15} - \frac{208396939998445}{5524740587274359428} a^{14} + \frac{165366173473718473}{11049481174548718856} a^{13} + \frac{2463125511084378957}{11049481174548718856} a^{12} + \frac{2000014930807322721}{11049481174548718856} a^{11} - \frac{2167931547845914127}{11049481174548718856} a^{10} + \frac{1547111559986500911}{11049481174548718856} a^{9} - \frac{354548078873674337}{2762370293637179714} a^{8} + \frac{2517439211178928029}{11049481174548718856} a^{7} + \frac{2465534358600718233}{5524740587274359428} a^{6} + \frac{750912707301535647}{5524740587274359428} a^{5} - \frac{2023068433457531431}{11049481174548718856} a^{4} + \frac{5196267876943131549}{11049481174548718856} a^{3} + \frac{499262387749515983}{1381185146818589857} a^{2} + \frac{425607944570129067}{1381185146818589857} a - \frac{368025787301650747}{1381185146818589857}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 51289.5294495 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.313788397.1, 8.0.69347235737.1, 8.4.901514064581.1

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

Sibling fields

Degree 16 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$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}$
17Data not computed