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Properties

Label 16.4.44499647686...0625.1
Degree $16$
Signature $[4, 6]$
Discriminant $3^{12}\cdot 5^{8}\cdot 11^{8}$
Root discriminant $16.91$
Ramified primes $3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times D_8$ (as 16T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 18, 9, -45, -84, -111, 99, 171, -23, -108, -29, 51, 18, -9, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 9*x^13 + 18*x^12 + 51*x^11 - 29*x^10 - 108*x^9 - 23*x^8 + 171*x^7 + 99*x^6 - 111*x^5 - 84*x^4 - 45*x^3 + 9*x^2 + 18*x + 9)
 
gp: K = bnfinit(x^16 - 8*x^14 - 9*x^13 + 18*x^12 + 51*x^11 - 29*x^10 - 108*x^9 - 23*x^8 + 171*x^7 + 99*x^6 - 111*x^5 - 84*x^4 - 45*x^3 + 9*x^2 + 18*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 9 x^{13} + 18 x^{12} + 51 x^{11} - 29 x^{10} - 108 x^{9} - 23 x^{8} + 171 x^{7} + 99 x^{6} - 111 x^{5} - 84 x^{4} - 45 x^{3} + 9 x^{2} + 18 x + 9 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(44499647686531640625=3^{12}\cdot 5^{8}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.91$
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:  $3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5}$, $\frac{1}{345} a^{14} - \frac{22}{345} a^{13} - \frac{8}{69} a^{12} - \frac{82}{345} a^{11} + \frac{37}{345} a^{10} - \frac{27}{115} a^{9} - \frac{89}{345} a^{8} - \frac{169}{345} a^{7} - \frac{61}{345} a^{6} + \frac{167}{345} a^{5} - \frac{44}{345} a^{4} + \frac{1}{23} a^{3} - \frac{9}{23} a^{2} + \frac{1}{23} a - \frac{2}{115}$, $\frac{1}{7645978659825} a^{15} - \frac{296115592}{2548659553275} a^{14} + \frac{740656696963}{7645978659825} a^{13} + \frac{56583131432}{849553184425} a^{12} - \frac{22764806036}{169910636885} a^{11} - \frac{41389091341}{849553184425} a^{10} + \frac{442781414486}{1529195731965} a^{9} - \frac{1046248247536}{2548659553275} a^{8} + \frac{75153932006}{1529195731965} a^{7} - \frac{3449746738}{12679898275} a^{6} + \frac{263772745427}{849553184425} a^{5} - \frac{4412954488}{12679898275} a^{4} - \frac{22743614153}{101946382131} a^{3} - \frac{4641697066}{169910636885} a^{2} - \frac{206891870519}{849553184425} a + \frac{290719203846}{849553184425}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:  \( 3619.65458774 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_8$ (as 16T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2\times D_8$
Character table for $C_2\times D_8$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{165}) \), 4.2.2475.1, 4.2.275.1, \(\Q(\sqrt{5}, \sqrt{33})\), 8.2.606436875.2, 8.2.606436875.1, 8.4.741200625.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$