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Properties

Label 16.8.29335496314...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 13^{4}\cdot 29^{10}$
Root discriminant $52.08$
Ramified primes $5, 13, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $OD_{16}:C_2^2$ (as 16T106)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![449591, -77161, -234549, 169226, -286394, 22608, 158548, -38297, -9708, -1294, -343, 1444, -554, 72, 4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 4*x^14 + 72*x^13 - 554*x^12 + 1444*x^11 - 343*x^10 - 1294*x^9 - 9708*x^8 - 38297*x^7 + 158548*x^6 + 22608*x^5 - 286394*x^4 + 169226*x^3 - 234549*x^2 - 77161*x + 449591)
 
gp: K = bnfinit(x^16 - 3*x^15 + 4*x^14 + 72*x^13 - 554*x^12 + 1444*x^11 - 343*x^10 - 1294*x^9 - 9708*x^8 - 38297*x^7 + 158548*x^6 + 22608*x^5 - 286394*x^4 + 169226*x^3 - 234549*x^2 - 77161*x + 449591, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 4 x^{14} + 72 x^{13} - 554 x^{12} + 1444 x^{11} - 343 x^{10} - 1294 x^{9} - 9708 x^{8} - 38297 x^{7} + 158548 x^{6} + 22608 x^{5} - 286394 x^{4} + 169226 x^{3} - 234549 x^{2} - 77161 x + 449591 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2933549631417734560791015625=5^{12}\cdot 13^{4}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.08$
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:  $5, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{139} a^{14} + \frac{51}{139} a^{13} + \frac{68}{139} a^{12} - \frac{6}{139} a^{11} - \frac{40}{139} a^{10} - \frac{5}{139} a^{9} - \frac{43}{139} a^{8} - \frac{35}{139} a^{7} - \frac{39}{139} a^{6} - \frac{46}{139} a^{5} - \frac{68}{139} a^{4} + \frac{62}{139} a^{3} - \frac{46}{139} a^{2} - \frac{38}{139} a + \frac{7}{139}$, $\frac{1}{489296472157484286812170599491583633631947179} a^{15} - \frac{763338319084774041889340132217232521351625}{489296472157484286812170599491583633631947179} a^{14} - \frac{162719095583391617122204640743399794797899773}{489296472157484286812170599491583633631947179} a^{13} - \frac{205847029997390248324011340003188776144806451}{489296472157484286812170599491583633631947179} a^{12} + \frac{203641067774240983118142506169476924645927344}{489296472157484286812170599491583633631947179} a^{11} - \frac{214373179900616800790755016707569760943709588}{489296472157484286812170599491583633631947179} a^{10} - \frac{231273439247645303120483482580595686189332134}{489296472157484286812170599491583633631947179} a^{9} + \frac{220406536554259446296756892911915409482557531}{489296472157484286812170599491583633631947179} a^{8} - \frac{63313863474588581895494697941919031796245340}{489296472157484286812170599491583633631947179} a^{7} + \frac{20007300100678535322709999461095210075525850}{489296472157484286812170599491583633631947179} a^{6} - \frac{127851203893549632508543481200498154532907266}{489296472157484286812170599491583633631947179} a^{5} + \frac{206514408308170299356374926620772690949453022}{489296472157484286812170599491583633631947179} a^{4} + \frac{219809928457859587625013635369089876132568642}{489296472157484286812170599491583633631947179} a^{3} - \frac{37876854323100260375610104296811399685884991}{489296472157484286812170599491583633631947179} a^{2} + \frac{54692381965047906481428990931031445554877963}{489296472157484286812170599491583633631947179} a + \frac{189982962172455049233034864670412830857564724}{489296472157484286812170599491583633631947179}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 20771882.9997 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}:C_2^2$ (as 16T106):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $OD_{16}:C_2^2$
Character table for $OD_{16}:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{145}) \), 4.4.725.1 x2, 4.4.4205.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{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}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$