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Properties

Label 16.0.52264543887360000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{12}\cdot 5^{4}\cdot 7^{4}$
Root discriminant $11.09$
Ramified primes $2, 3, 5, 7$
Class number $1$
Class group Trivial
Galois group $(C_2\times D_4):C_2^2$ (as 16T115)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 4, -12, 23, -36, 58, -102, 163, -222, 242, -198, 119, -54, 20, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 20*x^14 - 54*x^13 + 119*x^12 - 198*x^11 + 242*x^10 - 222*x^9 + 163*x^8 - 102*x^7 + 58*x^6 - 36*x^5 + 23*x^4 - 12*x^3 + 4*x^2 + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 20*x^14 - 54*x^13 + 119*x^12 - 198*x^11 + 242*x^10 - 222*x^9 + 163*x^8 - 102*x^7 + 58*x^6 - 36*x^5 + 23*x^4 - 12*x^3 + 4*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 20 x^{14} - 54 x^{13} + 119 x^{12} - 198 x^{11} + 242 x^{10} - 222 x^{9} + 163 x^{8} - 102 x^{7} + 58 x^{6} - 36 x^{5} + 23 x^{4} - 12 x^{3} + 4 x^{2} + 1 \)

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:  \(52264543887360000=2^{16}\cdot 3^{12}\cdot 5^{4}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.09$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{3}$, $\frac{1}{130} a^{14} - \frac{6}{65} a^{13} - \frac{27}{130} a^{12} - \frac{12}{65} a^{11} + \frac{31}{130} a^{10} + \frac{1}{65} a^{9} - \frac{7}{65} a^{8} + \frac{7}{65} a^{7} - \frac{1}{13} a^{6} - \frac{9}{65} a^{5} - \frac{9}{130} a^{4} - \frac{5}{13} a^{3} + \frac{29}{130} a^{2} + \frac{22}{65} a + \frac{59}{130}$, $\frac{1}{130} a^{15} + \frac{12}{65} a^{13} - \frac{23}{130} a^{12} + \frac{3}{130} a^{11} - \frac{8}{65} a^{10} + \frac{1}{13} a^{9} - \frac{12}{65} a^{8} + \frac{14}{65} a^{7} - \frac{4}{65} a^{6} + \frac{7}{26} a^{5} - \frac{14}{65} a^{4} + \frac{7}{65} a^{3} - \frac{63}{130} a^{2} - \frac{63}{130} a + \frac{29}{65}$

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:  \( \frac{163}{130} a^{15} - \frac{823}{130} a^{14} + \frac{2543}{130} a^{13} - \frac{6683}{130} a^{12} + \frac{536}{5} a^{11} - \frac{21361}{130} a^{10} + \frac{23839}{130} a^{9} - \frac{4081}{26} a^{8} + \frac{14557}{130} a^{7} - \frac{8869}{130} a^{6} + \frac{2492}{65} a^{5} - \frac{1536}{65} a^{4} + \frac{916}{65} a^{3} - \frac{428}{65} a^{2} + \frac{59}{130} a - \frac{19}{65} \) (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:  \( 248.17999333 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4):C_2^2$ (as 16T115):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 25 conjugacy class representatives for $(C_2\times D_4):C_2^2$
Character table for $(C_2\times D_4):C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.0.189.1, \(\Q(\zeta_{12})\), 4.0.3024.2, 8.0.9144576.3, 8.0.4665600.1, 8.0.25401600.1

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 R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
3Data not computed
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$