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Properties

Label 16.0.92475143538...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 109^{10}$
Root discriminant $41.96$
Ramified primes $5, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![531441, 0, 501552, 0, 243976, 0, 86927, 0, 20250, 0, 2818, 0, 276, 0, 23, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 23*x^14 + 276*x^12 + 2818*x^10 + 20250*x^8 + 86927*x^6 + 243976*x^4 + 501552*x^2 + 531441)
 
gp: K = bnfinit(x^16 + 23*x^14 + 276*x^12 + 2818*x^10 + 20250*x^8 + 86927*x^6 + 243976*x^4 + 501552*x^2 + 531441, 1)
 

Normalized defining polynomial

\( x^{16} + 23 x^{14} + 276 x^{12} + 2818 x^{10} + 20250 x^{8} + 86927 x^{6} + 243976 x^{4} + 501552 x^{2} + 531441 \)

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:  \(92475143538754579453515625=5^{8}\cdot 109^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.96$
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:  $5, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{30} a^{8} - \frac{1}{6} a^{6} + \frac{7}{30} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{1}{5}$, $\frac{1}{30} a^{9} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{11}{30} a - \frac{1}{2}$, $\frac{1}{60} a^{10} - \frac{1}{20} a^{6} - \frac{1}{2} a^{3} - \frac{19}{60} a^{2} + \frac{1}{4}$, $\frac{1}{120} a^{11} - \frac{1}{120} a^{10} - \frac{1}{60} a^{9} - \frac{1}{60} a^{8} + \frac{7}{120} a^{7} + \frac{13}{120} a^{6} - \frac{7}{60} a^{5} + \frac{23}{60} a^{4} - \frac{13}{40} a^{3} + \frac{59}{120} a^{2} + \frac{1}{40} a + \frac{11}{40}$, $\frac{1}{4800} a^{12} - \frac{11}{4800} a^{10} + \frac{61}{4800} a^{8} + \frac{823}{4800} a^{6} - \frac{687}{1600} a^{4} + \frac{309}{800} a^{2} + \frac{423}{1600}$, $\frac{1}{86400} a^{13} - \frac{1}{9600} a^{12} + \frac{149}{86400} a^{11} + \frac{11}{9600} a^{10} - \frac{193}{28800} a^{9} - \frac{61}{9600} a^{8} + \frac{343}{86400} a^{7} - \frac{823}{9600} a^{6} + \frac{1051}{9600} a^{5} - \frac{913}{3200} a^{4} + \frac{9007}{43200} a^{3} - \frac{309}{1600} a^{2} + \frac{39829}{86400} a + \frac{1177}{3200}$, $\frac{1}{19889452800} a^{14} - \frac{128939}{2486181600} a^{12} + \frac{8681441}{1657454400} a^{10} - \frac{62169169}{9944726400} a^{8} + \frac{4153997}{30693600} a^{6} - \frac{1165847237}{3977890560} a^{4} - \frac{1}{2} a^{3} + \frac{480470147}{3977890560} a^{2} - \frac{1}{2} a - \frac{8588761}{27283200}$, $\frac{1}{1074030451200} a^{15} - \frac{1}{39778905600} a^{14} - \frac{128939}{134253806400} a^{13} + \frac{128939}{4972363200} a^{12} + \frac{174426881}{89502537600} a^{11} - \frac{8681441}{3314908800} a^{10} + \frac{2258266991}{537015225600} a^{9} - \frac{269321711}{19889452800} a^{8} + \frac{35870717}{1657454400} a^{7} - \frac{14385197}{61387200} a^{6} + \frac{19253990971}{214806090240} a^{5} + \frac{2226618053}{7955781120} a^{4} - \frac{91276205437}{214806090240} a^{3} + \frac{182511613}{7955781120} a^{2} - \frac{11317081}{1473292800} a + \frac{3132121}{54566400}$

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:  $7$
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:  \( 6889381.56239 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{109}) \), \(\Q(\sqrt{545}) \), 4.0.2725.1 x2, 4.0.59405.1 x2, \(\Q(\sqrt{5}, \sqrt{109})\), 8.4.9616399718125.1 x2, 8.0.88223850625.1, 8.0.88223850625.2 x2

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 8 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.4.2.1$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}$
109Data not computed