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Properties

Label 16.0.20253399449...6336.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{34}\cdot 13^{2}\cdot 17^{8}$
Root discriminant $24.78$
Ramified primes $2, 13, 17$
Class number $2$
Class group $[2]$
Galois group $C_2.C_2\wr C_2^2$ (as 16T400)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![356, -864, 1938, -2720, 3287, -2814, 2161, -990, 391, 48, -106, 88, -31, 6, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 + 6*x^13 - 31*x^12 + 88*x^11 - 106*x^10 + 48*x^9 + 391*x^8 - 990*x^7 + 2161*x^6 - 2814*x^5 + 3287*x^4 - 2720*x^3 + 1938*x^2 - 864*x + 356)
 
gp: K = bnfinit(x^16 - 2*x^15 + 3*x^14 + 6*x^13 - 31*x^12 + 88*x^11 - 106*x^10 + 48*x^9 + 391*x^8 - 990*x^7 + 2161*x^6 - 2814*x^5 + 3287*x^4 - 2720*x^3 + 1938*x^2 - 864*x + 356, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 3 x^{14} + 6 x^{13} - 31 x^{12} + 88 x^{11} - 106 x^{10} + 48 x^{9} + 391 x^{8} - 990 x^{7} + 2161 x^{6} - 2814 x^{5} + 3287 x^{4} - 2720 x^{3} + 1938 x^{2} - 864 x + 356 \)

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:  \(20253399449972387086336=2^{34}\cdot 13^{2}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.78$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4324} a^{14} + \frac{109}{1081} a^{13} - \frac{198}{1081} a^{12} - \frac{387}{2162} a^{11} + \frac{511}{4324} a^{10} - \frac{501}{1081} a^{9} - \frac{491}{4324} a^{8} - \frac{629}{2162} a^{7} + \frac{615}{2162} a^{6} - \frac{214}{1081} a^{5} - \frac{73}{188} a^{4} - \frac{424}{1081} a^{3} + \frac{139}{1081} a^{2} - \frac{237}{1081} a + \frac{155}{1081}$, $\frac{1}{356729627556584} a^{15} - \frac{21107217615}{356729627556584} a^{14} + \frac{5647052880799}{178364813778292} a^{13} - \frac{10900822268288}{44591203444573} a^{12} - \frac{2411319796085}{15509983806808} a^{11} + \frac{70768334538639}{356729627556584} a^{10} + \frac{156928121398427}{356729627556584} a^{9} - \frac{49470510528679}{356729627556584} a^{8} + \frac{36139662842215}{178364813778292} a^{7} - \frac{12165637778111}{89182406889146} a^{6} + \frac{143347891696745}{356729627556584} a^{5} + \frac{63879149668045}{356729627556584} a^{4} - \frac{76318343297917}{178364813778292} a^{3} + \frac{23884342863309}{178364813778292} a^{2} + \frac{33698611111511}{89182406889146} a + \frac{40130274168401}{89182406889146}$

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:  \( 75344.014109 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.C_2\wr C_2^2$ (as 16T400):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 20 conjugacy class representatives for $C_2.C_2\wr C_2^2$
Character table for $C_2.C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.0.1088.2 x2, 4.0.2312.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.0.342102016.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.4$x^{4} + 12 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
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}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$