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Properties

Label 16.0.90546958151...6832.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 13^{6}\cdot 17^{15}$
Root discriminant $74.53$
Ramified primes $2, 13, 17$
Class number $41760$ (GRH)
Class group $[2, 2, 2, 6, 870]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![82055753, 0, 119927639, 0, 66518569, 0, 18002218, 0, 2536859, 0, 186524, 0, 7174, 0, 136, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 136*x^14 + 7174*x^12 + 186524*x^10 + 2536859*x^8 + 18002218*x^6 + 66518569*x^4 + 119927639*x^2 + 82055753)
 
gp: K = bnfinit(x^16 + 136*x^14 + 7174*x^12 + 186524*x^10 + 2536859*x^8 + 18002218*x^6 + 66518569*x^4 + 119927639*x^2 + 82055753, 1)
 

Normalized defining polynomial

\( x^{16} + 136 x^{14} + 7174 x^{12} + 186524 x^{10} + 2536859 x^{8} + 18002218 x^{6} + 66518569 x^{4} + 119927639 x^{2} + 82055753 \)

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:  \(905469581514181342527129976832=2^{16}\cdot 13^{6}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.53$
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:  $2, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13} a^{6} + \frac{6}{13} a^{4} - \frac{2}{13} a^{2}$, $\frac{1}{13} a^{7} + \frac{6}{13} a^{5} - \frac{2}{13} a^{3}$, $\frac{1}{169} a^{8} + \frac{6}{169} a^{6} - \frac{28}{169} a^{4} + \frac{3}{13} a^{2}$, $\frac{1}{169} a^{9} + \frac{6}{169} a^{7} - \frac{28}{169} a^{5} + \frac{3}{13} a^{3}$, $\frac{1}{2197} a^{10} + \frac{6}{2197} a^{8} - \frac{28}{2197} a^{6} + \frac{3}{169} a^{4} + \frac{3}{13} a^{2}$, $\frac{1}{2197} a^{11} + \frac{6}{2197} a^{9} - \frac{28}{2197} a^{7} + \frac{3}{169} a^{5} + \frac{3}{13} a^{3}$, $\frac{1}{28561} a^{12} + \frac{6}{28561} a^{10} - \frac{28}{28561} a^{8} + \frac{3}{2197} a^{6} + \frac{16}{169} a^{4} - \frac{2}{13} a^{2}$, $\frac{1}{28561} a^{13} + \frac{6}{28561} a^{11} - \frac{28}{28561} a^{9} + \frac{3}{2197} a^{7} + \frac{16}{169} a^{5} - \frac{2}{13} a^{3}$, $\frac{1}{65722530488719} a^{14} + \frac{839039025}{65722530488719} a^{12} + \frac{9540465803}{65722530488719} a^{10} + \frac{2617297649}{5055579268363} a^{8} - \frac{4729657454}{388890712951} a^{6} - \frac{14627075999}{29914670227} a^{4} + \frac{895106165}{2301128479} a^{2} + \frac{71505139}{177009883}$, $\frac{1}{65722530488719} a^{15} + \frac{839039025}{65722530488719} a^{13} + \frac{9540465803}{65722530488719} a^{11} + \frac{2617297649}{5055579268363} a^{9} - \frac{4729657454}{388890712951} a^{7} - \frac{14627075999}{29914670227} a^{5} + \frac{895106165}{2301128479} a^{3} + \frac{71505139}{177009883} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{870}$, which has order $41760$ (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:  \( 3640.01221338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

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 $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 R $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
17Data not computed