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Properties

Label 16.0.14958734309...296.10
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{12}$
Root discriminant $18.24$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_2\wr C_2^2$ (as 16T149)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, -36, 0, 150, -288, 240, -48, -47, 0, 32, -24, 18, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 18*x^12 - 24*x^11 + 32*x^10 - 47*x^8 - 48*x^7 + 240*x^6 - 288*x^5 + 150*x^4 - 36*x^2 + 9)
 
gp: K = bnfinit(x^16 - 4*x^14 + 18*x^12 - 24*x^11 + 32*x^10 - 47*x^8 - 48*x^7 + 240*x^6 - 288*x^5 + 150*x^4 - 36*x^2 + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} + 18 x^{12} - 24 x^{11} + 32 x^{10} - 47 x^{8} - 48 x^{7} + 240 x^{6} - 288 x^{5} + 150 x^{4} - 36 x^{2} + 9 \)

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:  \(149587343098087735296=2^{48}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.24$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5}$, $\frac{1}{555} a^{14} + \frac{92}{555} a^{13} + \frac{9}{185} a^{12} + \frac{133}{555} a^{11} - \frac{97}{555} a^{10} + \frac{61}{185} a^{9} + \frac{149}{555} a^{8} + \frac{49}{555} a^{7} + \frac{8}{185} a^{6} - \frac{172}{555} a^{5} + \frac{139}{555} a^{4} - \frac{63}{185} a^{3} - \frac{4}{37} a^{2} - \frac{31}{185} a + \frac{36}{185}$, $\frac{1}{2139916772835} a^{15} - \frac{59555512}{713305590945} a^{14} + \frac{176169622721}{2139916772835} a^{13} + \frac{118720897774}{713305590945} a^{12} - \frac{123725787682}{713305590945} a^{11} + \frac{299466418753}{713305590945} a^{10} - \frac{32049526691}{427983354567} a^{9} + \frac{65792916799}{713305590945} a^{8} - \frac{872428285478}{2139916772835} a^{7} + \frac{320628279457}{713305590945} a^{6} + \frac{10205485445}{47553706063} a^{5} + \frac{21157361986}{237768530315} a^{4} + \frac{189110877964}{713305590945} a^{3} + \frac{39192112728}{237768530315} a^{2} + \frac{24077439413}{237768530315} a - \frac{69136013241}{237768530315}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{241312224}{1285235299} a^{15} - \frac{645035204}{3855705897} a^{14} + \frac{2245506176}{3855705897} a^{13} + \frac{2023422806}{3855705897} a^{12} - \frac{10905180728}{3855705897} a^{11} + \frac{2528883144}{1285235299} a^{10} - \frac{5940389760}{1285235299} a^{9} - \frac{13704756589}{3855705897} a^{8} + \frac{19200803248}{3855705897} a^{7} + \frac{52327428760}{3855705897} a^{6} - \frac{123204236704}{3855705897} a^{5} + \frac{33956987826}{1285235299} a^{4} - \frac{12145538352}{1285235299} a^{3} - \frac{2473355156}{1285235299} a^{2} + \frac{1958669888}{1285235299} a + \frac{2472211820}{1285235299} \) (order $6$)
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:  \( 35379.2542371 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T149):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), 4.0.27648.1, 4.4.27648.1, \(\Q(\sqrt{-2}, \sqrt{-3})\), 8.0.764411904.5, 8.4.12230590464.4 x2, 8.0.254803968.2 x2

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

Sibling fields

Degree 8 siblings: data not computed
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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$