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Properties

Label 16.0.10737418240...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{10}$
Root discriminant $15.47$
Ramified primes $2, 5$
Class number $2$
Class group $[2]$
Galois group $C_2\times C_4\wr C_2$ (as 16T111)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 64, -172, 214, -40, 0, -244, 308, -132, 64, -68, 26, 0, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 26*x^12 - 68*x^11 + 64*x^10 - 132*x^9 + 308*x^8 - 244*x^7 - 40*x^5 + 214*x^4 - 172*x^3 + 64*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 + 26*x^12 - 68*x^11 + 64*x^10 - 132*x^9 + 308*x^8 - 244*x^7 - 40*x^5 + 214*x^4 - 172*x^3 + 64*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} + 26 x^{12} - 68 x^{11} + 64 x^{10} - 132 x^{9} + 308 x^{8} - 244 x^{7} - 40 x^{5} + 214 x^{4} - 172 x^{3} + 64 x^{2} - 12 x + 1 \)

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:  \(10737418240000000000=2^{40}\cdot 5^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.47$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{146701938568763} a^{15} + \frac{63299762640378}{146701938568763} a^{14} - \frac{17701858863508}{146701938568763} a^{13} + \frac{923810181840}{146701938568763} a^{12} + \frac{32671466895063}{146701938568763} a^{11} - \frac{10681159853183}{146701938568763} a^{10} - \frac{4174769296131}{146701938568763} a^{9} - \frac{50549735140323}{146701938568763} a^{8} - \frac{70343014979494}{146701938568763} a^{7} - \frac{33652397649916}{146701938568763} a^{6} - \frac{783520386462}{146701938568763} a^{5} + \frac{6353121380427}{146701938568763} a^{4} + \frac{13651680823186}{146701938568763} a^{3} - \frac{30304166937644}{146701938568763} a^{2} + \frac{17299372555650}{146701938568763} a + \frac{64737269196537}{146701938568763}$

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:  \( \frac{441864478}{907825879} a^{15} - \frac{1009343044}{907825879} a^{14} - \frac{915745672}{907825879} a^{13} + \frac{1764179055}{907825879} a^{12} + \frac{12417868758}{907825879} a^{11} - \frac{10001777090}{907825879} a^{10} - \frac{14073230298}{907825879} a^{9} - \frac{30125656986}{907825879} a^{8} + \frac{50924508486}{907825879} a^{7} + \frac{84873057314}{907825879} a^{6} - \frac{92921581328}{907825879} a^{5} - \frac{69076283695}{907825879} a^{4} + \frac{45932144306}{907825879} a^{3} + \frac{67011760102}{907825879} a^{2} - \frac{35865659754}{907825879} a + \frac{6296575995}{907825879} \) (order $4$)
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:  \( 1495.7495683 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4\wr C_2$ (as 16T111):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $C_2\times C_4\wr C_2$
Character table for $C_2\times C_4\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{10}) \), 4.0.320.1, 4.0.1280.1, \(\Q(i, \sqrt{10})\), 8.0.8192000.1, 8.0.819200000.1, 8.0.163840000.2

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

Sibling fields

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$