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Properties

Label 16.0.19411551302...7264.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 7^{10}$
Root discriminant $16.05$
Ramified primes $2, 7$
Class number $2$
Class group $[2]$
Galois group $D_4:D_4$ (as 16T141)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -32, 304, -108, 512, -344, 460, -340, 258, -224, 88, -60, 40, -8, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 - 8*x^13 + 40*x^12 - 60*x^11 + 88*x^10 - 224*x^9 + 258*x^8 - 340*x^7 + 460*x^6 - 344*x^5 + 512*x^4 - 108*x^3 + 304*x^2 - 32*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 - 8*x^13 + 40*x^12 - 60*x^11 + 88*x^10 - 224*x^9 + 258*x^8 - 340*x^7 + 460*x^6 - 344*x^5 + 512*x^4 - 108*x^3 + 304*x^2 - 32*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} - 8 x^{13} + 40 x^{12} - 60 x^{11} + 88 x^{10} - 224 x^{9} + 258 x^{8} - 340 x^{7} + 460 x^{6} - 344 x^{5} + 512 x^{4} - 108 x^{3} + 304 x^{2} - 32 x + 1 \)

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:  \(19411551302151307264=2^{36}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.05$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{1}{8} a^{4} + \frac{3}{8} a^{2} + \frac{3}{8}$, $\frac{1}{192606532136} a^{15} - \frac{5912285281}{192606532136} a^{14} - \frac{401992835}{11329796008} a^{13} + \frac{13340571713}{192606532136} a^{12} - \frac{10629440461}{192606532136} a^{11} - \frac{21364101717}{192606532136} a^{10} + \frac{2491970785}{192606532136} a^{9} + \frac{8427863005}{192606532136} a^{8} + \frac{80795265065}{192606532136} a^{7} - \frac{1055288977}{4697720296} a^{6} + \frac{42466966069}{192606532136} a^{5} - \frac{81377062307}{192606532136} a^{4} - \frac{4634130801}{11329796008} a^{3} - \frac{43177231641}{192606532136} a^{2} - \frac{28395263435}{192606532136} a - \frac{72285460547}{192606532136}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{6959868183}{96303266068} a^{15} + \frac{26772332131}{96303266068} a^{14} - \frac{1620652545}{5664898004} a^{13} + \frac{17810629862}{24075816517} a^{12} - \frac{74532240031}{24075816517} a^{11} + \frac{403642037763}{96303266068} a^{10} - \frac{350536769651}{48151633034} a^{9} + \frac{453255779260}{24075816517} a^{8} - \frac{1999215375709}{96303266068} a^{7} + \frac{71601316283}{2348860148} a^{6} - \frac{4095834278255}{96303266068} a^{5} + \frac{814746719624}{24075816517} a^{4} - \frac{139065817533}{2832449002} a^{3} + \frac{1263244423595}{96303266068} a^{2} - \frac{790564416346}{24075816517} a + \frac{58782291560}{24075816517} \) (order $8$)
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:  \( 2499.35245563 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:D_4$ (as 16T141):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $D_4:D_4$
Character table for $D_4:D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.2.448.1, 4.2.1792.1, \(\Q(\zeta_{8})\), 8.2.1101463552.1 x2, 8.2.4405854208.1 x2, 8.0.3211264.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$