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Properties

Label 16.4.36555024820...4401.1
Degree $16$
Signature $[4, 6]$
Discriminant $17^{12}\cdot 89^{4}$
Root discriminant $25.71$
Ramified primes $17, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\wr C_4$ (as 16T157)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-404, 1912, -4385, 6535, -7289, 6826, -5678, 4186, -2608, 1288, -432, 30, 74, -58, 25, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 25*x^14 - 58*x^13 + 74*x^12 + 30*x^11 - 432*x^10 + 1288*x^9 - 2608*x^8 + 4186*x^7 - 5678*x^6 + 6826*x^5 - 7289*x^4 + 6535*x^3 - 4385*x^2 + 1912*x - 404)
 
gp: K = bnfinit(x^16 - 7*x^15 + 25*x^14 - 58*x^13 + 74*x^12 + 30*x^11 - 432*x^10 + 1288*x^9 - 2608*x^8 + 4186*x^7 - 5678*x^6 + 6826*x^5 - 7289*x^4 + 6535*x^3 - 4385*x^2 + 1912*x - 404, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 25 x^{14} - 58 x^{13} + 74 x^{12} + 30 x^{11} - 432 x^{10} + 1288 x^{9} - 2608 x^{8} + 4186 x^{7} - 5678 x^{6} + 6826 x^{5} - 7289 x^{4} + 6535 x^{3} - 4385 x^{2} + 1912 x - 404 \)

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:  \(36555024820228837034401=17^{12}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.71$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{32} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} + \frac{1}{32} a^{8} + \frac{1}{8} a^{6} - \frac{1}{32} a^{5} + \frac{3}{16} a^{4} - \frac{7}{16} a^{3} + \frac{15}{32} a^{2} + \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{32} a^{12} - \frac{1}{16} a^{10} + \frac{1}{32} a^{9} - \frac{1}{16} a^{8} + \frac{7}{32} a^{6} + \frac{1}{8} a^{5} + \frac{1}{16} a^{4} + \frac{7}{32} a^{3} + \frac{3}{16} a^{2} - \frac{1}{2}$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{12} - \frac{1}{64} a^{11} + \frac{1}{64} a^{10} + \frac{3}{64} a^{9} - \frac{1}{64} a^{8} + \frac{3}{64} a^{7} - \frac{15}{64} a^{6} - \frac{11}{64} a^{5} + \frac{11}{64} a^{4} + \frac{1}{64} a^{3} + \frac{5}{64} a^{2} + \frac{3}{8} a - \frac{7}{16}$, $\frac{1}{256} a^{14} - \frac{1}{128} a^{13} - \frac{1}{64} a^{12} - \frac{1}{64} a^{11} - \frac{5}{128} a^{10} - \frac{3}{64} a^{9} + \frac{3}{128} a^{8} - \frac{9}{128} a^{7} - \frac{1}{16} a^{6} - \frac{13}{64} a^{5} - \frac{27}{128} a^{4} + \frac{27}{64} a^{3} - \frac{31}{256} a^{2} - \frac{27}{64} a + \frac{5}{64}$, $\frac{1}{351488} a^{15} + \frac{125}{351488} a^{14} + \frac{711}{175744} a^{13} + \frac{229}{21968} a^{12} + \frac{213}{175744} a^{11} - \frac{10313}{175744} a^{10} + \frac{1857}{175744} a^{9} + \frac{2403}{43936} a^{8} - \frac{21767}{175744} a^{7} + \frac{9507}{87872} a^{6} - \frac{89}{175744} a^{5} + \frac{2649}{175744} a^{4} + \frac{140101}{351488} a^{3} - \frac{118013}{351488} a^{2} - \frac{643}{2746} a + \frac{31819}{87872}$

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:  \( 539030.426161 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T157):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.191193684049.1, 8.4.2148243641.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
89Data not computed