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Properties

Label 16.0.95735899582...8944.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{54}\cdot 3^{12}$
Root discriminant $23.65$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![184, -128, -320, -32, 272, -224, 160, 32, 388, -144, 280, -48, 100, -8, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 - 8*x^13 + 100*x^12 - 48*x^11 + 280*x^10 - 144*x^9 + 388*x^8 + 32*x^7 + 160*x^6 - 224*x^5 + 272*x^4 - 32*x^3 - 320*x^2 - 128*x + 184)
 
gp: K = bnfinit(x^16 + 16*x^14 - 8*x^13 + 100*x^12 - 48*x^11 + 280*x^10 - 144*x^9 + 388*x^8 + 32*x^7 + 160*x^6 - 224*x^5 + 272*x^4 - 32*x^3 - 320*x^2 - 128*x + 184, 1)
 

Normalized defining polynomial

\( x^{16} + 16 x^{14} - 8 x^{13} + 100 x^{12} - 48 x^{11} + 280 x^{10} - 144 x^{9} + 388 x^{8} + 32 x^{7} + 160 x^{6} - 224 x^{5} + 272 x^{4} - 32 x^{3} - 320 x^{2} - 128 x + 184 \)

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:  \(9573589958277615058944=2^{54}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.65$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{28} a^{12} - \frac{1}{14} a^{11} - \frac{1}{14} a^{10} - \frac{1}{14} a^{9} + \frac{3}{14} a^{8} + \frac{1}{14} a^{7} - \frac{3}{14} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{28} a^{13} + \frac{1}{28} a^{11} - \frac{3}{14} a^{10} + \frac{1}{14} a^{9} - \frac{1}{14} a^{7} - \frac{1}{14} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{231196} a^{14} + \frac{3837}{231196} a^{13} - \frac{1020}{57799} a^{12} + \frac{3245}{57799} a^{11} - \frac{473}{2513} a^{10} + \frac{8278}{57799} a^{9} - \frac{1831}{57799} a^{8} - \frac{1163}{57799} a^{7} + \frac{5195}{115598} a^{6} + \frac{25433}{57799} a^{5} + \frac{14692}{57799} a^{4} - \frac{17978}{57799} a^{3} + \frac{9119}{57799} a^{2} + \frac{14317}{57799} a - \frac{1020}{2513}$, $\frac{1}{79957055836} a^{15} - \frac{44963}{39978527918} a^{14} + \frac{3689138}{19989263959} a^{13} + \frac{805910667}{79957055836} a^{12} - \frac{420580511}{11422436548} a^{11} + \frac{1499864657}{39978527918} a^{10} + \frac{5513955425}{39978527918} a^{9} - \frac{4899746393}{39978527918} a^{8} - \frac{1023245829}{19989263959} a^{7} - \frac{1478288125}{19989263959} a^{6} - \frac{4442972904}{19989263959} a^{5} + \frac{1390677775}{2855609137} a^{4} - \frac{4700696317}{19989263959} a^{3} - \frac{3065797459}{19989263959} a^{2} - \frac{7331940533}{19989263959} a + \frac{424030279}{869098433}$

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:  \( -1 \) (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:  \( 22843.7922823 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), 4.0.18432.2, \(\Q(\sqrt{2}, \sqrt{3})\), 4.0.2048.2, 4.0.27648.1 x2, 4.0.13824.1 x2, 8.0.1358954496.3, 8.0.3057647616.7, 8.8.12230590464.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed