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Properties

Label 16.0.52891851109...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{4}\cdot 41^{4}$
Root discriminant $30.39$
Ramified primes $2, 5, 13, 41$
Class number $40$ (GRH)
Class group $[40]$ (GRH)
Galois group $C_4^2:C_2^2$ (as 16T117)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![961, -2542, 6693, -12602, 18108, -17788, 16743, -11980, 8303, -4428, 2357, -1024, 420, -134, 39, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 39*x^14 - 134*x^13 + 420*x^12 - 1024*x^11 + 2357*x^10 - 4428*x^9 + 8303*x^8 - 11980*x^7 + 16743*x^6 - 17788*x^5 + 18108*x^4 - 12602*x^3 + 6693*x^2 - 2542*x + 961)
 
gp: K = bnfinit(x^16 - 6*x^15 + 39*x^14 - 134*x^13 + 420*x^12 - 1024*x^11 + 2357*x^10 - 4428*x^9 + 8303*x^8 - 11980*x^7 + 16743*x^6 - 17788*x^5 + 18108*x^4 - 12602*x^3 + 6693*x^2 - 2542*x + 961, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 39 x^{14} - 134 x^{13} + 420 x^{12} - 1024 x^{11} + 2357 x^{10} - 4428 x^{9} + 8303 x^{8} - 11980 x^{7} + 16743 x^{6} - 17788 x^{5} + 18108 x^{4} - 12602 x^{3} + 6693 x^{2} - 2542 x + 961 \)

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:  \(528918511098265600000000=2^{24}\cdot 5^{8}\cdot 13^{4}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.39$
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:  $2, 5, 13, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{587009} a^{14} - \frac{214792}{587009} a^{13} + \frac{230444}{587009} a^{12} + \frac{256689}{587009} a^{11} + \frac{181801}{587009} a^{10} - \frac{138340}{587009} a^{9} - \frac{147562}{587009} a^{8} - \frac{178935}{587009} a^{7} + \frac{237592}{587009} a^{6} + \frac{99508}{587009} a^{5} - \frac{73460}{587009} a^{4} + \frac{18035}{587009} a^{3} + \frac{140618}{587009} a^{2} - \frac{190728}{587009} a - \frac{9888}{587009}$, $\frac{1}{6002155921415869405197311} a^{15} - \frac{3348632243739632851}{6002155921415869405197311} a^{14} + \frac{31935237238429125860843}{6002155921415869405197311} a^{13} - \frac{848453654378730835132934}{6002155921415869405197311} a^{12} - \frac{227001914827066796722050}{6002155921415869405197311} a^{11} + \frac{2285872463984157760309924}{6002155921415869405197311} a^{10} + \frac{2295127176670014365417982}{6002155921415869405197311} a^{9} - \frac{2225152868804639651303816}{6002155921415869405197311} a^{8} + \frac{2733247506339327770724974}{6002155921415869405197311} a^{7} - \frac{506140552215086854288692}{6002155921415869405197311} a^{6} + \frac{526035317594861312449399}{6002155921415869405197311} a^{5} + \frac{555476030855739812439811}{6002155921415869405197311} a^{4} + \frac{1817494398999804520733310}{6002155921415869405197311} a^{3} - \frac{2062991773316532963781148}{6002155921415869405197311} a^{2} + \frac{1460858789628949622735006}{6002155921415869405197311} a + \frac{60900500415300210539213}{193617932948899013070881}$

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

Class group and class number

$C_{40}$, which has order $40$ (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:  \( 3710.59482488 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2$ (as 16T117):

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 $C_4^2:C_2^2$
Character table for $C_4^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.65600.5, \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.65600.2, 8.0.4303360000.3, 8.8.432640000.1, 8.0.727267840000.20

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$