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Properties

Label 16.0.10633333143...5936.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 223^{4}$
Root discriminant $13.39$
Ramified primes $2, 3, 223$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^2:S_4:C_2$ (as 16T724)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 3, -32, -15, 70, 32, -80, -53, 62, 66, -72, 0, 18, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 4*x^14 + 18*x^13 - 72*x^11 + 66*x^10 + 62*x^9 - 53*x^8 - 80*x^7 + 32*x^6 + 70*x^5 - 15*x^4 - 32*x^3 + 3*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 4*x^14 + 18*x^13 - 72*x^11 + 66*x^10 + 62*x^9 - 53*x^8 - 80*x^7 + 32*x^6 + 70*x^5 - 15*x^4 - 32*x^3 + 3*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 4 x^{14} + 18 x^{13} - 72 x^{11} + 66 x^{10} + 62 x^{9} - 53 x^{8} - 80 x^{7} + 32 x^{6} + 70 x^{5} - 15 x^{4} - 32 x^{3} + 3 x^{2} + 6 x + 1 \)

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:  \(1063333314324135936=2^{16}\cdot 3^{8}\cdot 223^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.39$
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:  $2, 3, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{71} a^{14} - \frac{30}{71} a^{13} + \frac{6}{71} a^{12} - \frac{29}{71} a^{11} + \frac{21}{71} a^{10} - \frac{20}{71} a^{9} + \frac{23}{71} a^{8} - \frac{28}{71} a^{7} + \frac{30}{71} a^{6} + \frac{26}{71} a^{5} + \frac{35}{71} a^{4} + \frac{17}{71} a^{3} - \frac{5}{71} a^{2} - \frac{15}{71} a + \frac{29}{71}$, $\frac{1}{145282259} a^{15} - \frac{214579}{145282259} a^{14} + \frac{54721304}{145282259} a^{13} - \frac{6221965}{145282259} a^{12} + \frac{48935045}{145282259} a^{11} - \frac{70112957}{145282259} a^{10} - \frac{8830933}{145282259} a^{9} - \frac{3439963}{145282259} a^{8} + \frac{10984573}{145282259} a^{7} - \frac{67384761}{145282259} a^{6} - \frac{66537632}{145282259} a^{5} - \frac{64183812}{145282259} a^{4} + \frac{14781422}{145282259} a^{3} - \frac{19534}{145282259} a^{2} - \frac{58470512}{145282259} a + \frac{31234342}{145282259}$

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:  \( \frac{115777750}{145282259} a^{15} - \frac{215713166}{145282259} a^{14} - \frac{456182335}{145282259} a^{13} + \frac{1959790577}{145282259} a^{12} + \frac{100799493}{145282259} a^{11} - \frac{7726678748}{145282259} a^{10} + \frac{6789358833}{145282259} a^{9} + \frac{5565628541}{145282259} a^{8} - \frac{3915828392}{145282259} a^{7} - \frac{7085921734}{145282259} a^{6} + \frac{2188951687}{145282259} a^{5} + \frac{5124134459}{145282259} a^{4} - \frac{1299726853}{145282259} a^{3} - \frac{1841387434}{145282259} a^{2} + \frac{581986273}{145282259} a + \frac{117995899}{145282259} \) (order $12$)
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:  \( 1356.52366954 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:S_4:C_2$ (as 16T724):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 28 conjugacy class representatives for $C_2\times C_2^2:S_4:C_2$
Character table for $C_2\times C_2^2:S_4:C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 8.4.1031180544.2

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
223Data not computed