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Properties

Label 16.0.59262062886...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 79^{8}$
Root discriminant $19.87$
Ramified primes $5, 79$
Class number $5$
Class group $[5]$
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -320, 624, -1080, 1488, -1030, 881, -280, 388, 0, 97, 0, 7, -10, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 10*x^13 + 7*x^12 + 97*x^10 + 388*x^8 - 280*x^7 + 881*x^6 - 1030*x^5 + 1488*x^4 - 1080*x^3 + 624*x^2 - 320*x + 256)
 
gp: K = bnfinit(x^16 + 2*x^14 - 10*x^13 + 7*x^12 + 97*x^10 + 388*x^8 - 280*x^7 + 881*x^6 - 1030*x^5 + 1488*x^4 - 1080*x^3 + 624*x^2 - 320*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} + 2 x^{14} - 10 x^{13} + 7 x^{12} + 97 x^{10} + 388 x^{8} - 280 x^{7} + 881 x^{6} - 1030 x^{5} + 1488 x^{4} - 1080 x^{3} + 624 x^{2} - 320 x + 256 \)

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:  \(592620628869750390625=5^{8}\cdot 79^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.87$
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:  $5, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{16} a^{5} - \frac{3}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{10} - \frac{1}{16} a^{8} - \frac{3}{16} a^{7} - \frac{5}{32} a^{6} + \frac{1}{16} a^{5} - \frac{3}{32} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{64} a^{13} + \frac{1}{64} a^{11} - \frac{1}{32} a^{9} - \frac{3}{32} a^{8} - \frac{5}{64} a^{7} + \frac{1}{32} a^{6} - \frac{3}{64} a^{5} + \frac{1}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{12} - \frac{1}{16} a^{10} - \frac{3}{32} a^{9} - \frac{1}{64} a^{8} + \frac{7}{32} a^{7} + \frac{7}{64} a^{6} - \frac{5}{32} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{5272250459431936} a^{15} - \frac{35086997524683}{5272250459431936} a^{14} + \frac{2542751559787}{5272250459431936} a^{13} - \frac{54918452568947}{5272250459431936} a^{12} - \frac{4359126684583}{659031307428992} a^{11} + \frac{40744589346203}{659031307428992} a^{10} + \frac{445096294637401}{5272250459431936} a^{9} - \frac{137015898364947}{5272250459431936} a^{8} - \frac{611943946095227}{5272250459431936} a^{7} + \frac{751707198247265}{5272250459431936} a^{6} - \frac{625129700738757}{2636125229715968} a^{5} - \frac{39846022485565}{659031307428992} a^{4} - \frac{239903240447867}{659031307428992} a^{3} - \frac{131148514860975}{329515653714496} a^{2} - \frac{30694114410031}{82378913428624} a + \frac{7283193052735}{20594728357156}$

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

Class group and class number

$C_{5}$, which has order $5$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5928.11603186 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-395}) \), \(\Q(\sqrt{-79}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-79})\), 4.0.31205.1 x2, 4.2.1975.1 x2, 8.0.24343800625.1, 8.0.4868760125.1 x4, 8.2.308149375.1 x4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$79$79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$