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Properties

Label 16.0.68158561146...0416.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 13^{4}\cdot 17^{12}$
Root discriminant $26.74$
Ramified primes $2, 13, 17$
Class number $2$
Class group $[2]$
Galois group $C_2^5.C_2$ (as 16T79)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2704, -5928, 8392, -9858, 11369, -10183, 7481, -4060, 1377, -71, -288, 187, -55, -2, 11, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 11*x^14 - 2*x^13 - 55*x^12 + 187*x^11 - 288*x^10 - 71*x^9 + 1377*x^8 - 4060*x^7 + 7481*x^6 - 10183*x^5 + 11369*x^4 - 9858*x^3 + 8392*x^2 - 5928*x + 2704)
 
gp: K = bnfinit(x^16 - 5*x^15 + 11*x^14 - 2*x^13 - 55*x^12 + 187*x^11 - 288*x^10 - 71*x^9 + 1377*x^8 - 4060*x^7 + 7481*x^6 - 10183*x^5 + 11369*x^4 - 9858*x^3 + 8392*x^2 - 5928*x + 2704, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 11 x^{14} - 2 x^{13} - 55 x^{12} + 187 x^{11} - 288 x^{10} - 71 x^{9} + 1377 x^{8} - 4060 x^{7} + 7481 x^{6} - 10183 x^{5} + 11369 x^{4} - 9858 x^{3} + 8392 x^{2} - 5928 x + 2704 \)

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:  \(68158561146958659260416=2^{12}\cdot 13^{4}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.74$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{11} - \frac{1}{4} a^{9} + \frac{1}{12} a^{8} + \frac{5}{12} a^{6} + \frac{1}{12} a^{5} + \frac{1}{6} a^{4} - \frac{1}{4} a^{3} - \frac{1}{12} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{4} a^{10} - \frac{1}{6} a^{9} + \frac{1}{12} a^{8} - \frac{1}{12} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{12} a^{4} - \frac{1}{3} a^{3} + \frac{5}{12} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{72} a^{14} + \frac{1}{36} a^{13} - \frac{1}{24} a^{12} + \frac{5}{72} a^{11} + \frac{2}{9} a^{10} - \frac{1}{8} a^{9} - \frac{7}{72} a^{8} - \frac{11}{72} a^{6} + \frac{1}{24} a^{5} - \frac{7}{36} a^{4} + \frac{3}{8} a^{3} - \frac{13}{36} a^{2} - \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{896897161067208764112} a^{15} - \frac{638332496543992711}{298965720355736254704} a^{14} + \frac{27085179341841932309}{896897161067208764112} a^{13} + \frac{321617416960316047}{448448580533604382056} a^{12} - \frac{17666733411870031087}{298965720355736254704} a^{11} + \frac{184894845090206998609}{896897161067208764112} a^{10} + \frac{54997672456260882887}{224224290266802191028} a^{9} - \frac{28008286776105581497}{896897161067208764112} a^{8} + \frac{214000131044955407671}{896897161067208764112} a^{7} + \frac{51082129333630838267}{112112145133401095514} a^{6} + \frac{255683321285830669003}{896897161067208764112} a^{5} - \frac{176409389475522152213}{896897161067208764112} a^{4} - \frac{259058766565328312303}{896897161067208764112} a^{3} - \frac{63804843539046605113}{224224290266802191028} a^{2} + \frac{25295303966492504267}{56056072566700547757} a - \frac{3115985182443438853}{8624011164107776578}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 112087.126914 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2$ (as 16T79):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $C_2^5.C_2$
Character table for $C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.30056.1, 4.4.4913.1, 4.0.39304.1, 4.0.3757.1, 4.4.510952.1, 4.0.63869.1, 4.0.2312.1, 8.0.261071946304.4, 8.8.261071946304.2, 8.0.261071946304.10, 8.0.261071946304.11, 8.0.4079249161.1, 8.0.1544804416.1, 8.0.903363136.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$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$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$