Properties

Label 16.0.27487790694...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{46}\cdot 5^{8}$
Root discriminant $16.40$
Ramified primes $2, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2\wr C_2$ (as 16T39)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 24*x^13 + 72*x^12 - 24*x^11 + 40*x^10 - 280*x^9 + 240*x^8 + 144*x^7 - 88*x^6 - 192*x^5 + 80*x^4 + 48*x^3 - 16*x + 4)
 
gp: K = bnfinit(x^16 - 4*x^14 - 24*x^13 + 72*x^12 - 24*x^11 + 40*x^10 - 280*x^9 + 240*x^8 + 144*x^7 - 88*x^6 - 192*x^5 + 80*x^4 + 48*x^3 - 16*x + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -16, 0, 48, 80, -192, -88, 144, 240, -280, 40, -24, 72, -24, -4, 0, 1]);
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 24 x^{13} + 72 x^{12} - 24 x^{11} + 40 x^{10} - 280 x^{9} + 240 x^{8} + 144 x^{7} - 88 x^{6} - 192 x^{5} + 80 x^{4} + 48 x^{3} - 16 x + 4 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(27487790694400000000=2^{46}\cdot 5^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $16.40$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{12} + \frac{1}{20} a^{11} + \frac{1}{10} a^{10} + \frac{1}{20} a^{9} + \frac{1}{10} a^{8} - \frac{1}{2} a^{7} + \frac{1}{10} a^{5} - \frac{3}{10} a^{4} - \frac{1}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{40} a^{13} - \frac{1}{10} a^{11} + \frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{1}{20} a^{8} - \frac{1}{5} a^{6} - \frac{9}{20} a^{5} + \frac{1}{10} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{3}{10} a + \frac{1}{10}$, $\frac{1}{520} a^{14} - \frac{1}{130} a^{13} - \frac{1}{65} a^{12} + \frac{9}{130} a^{11} + \frac{31}{260} a^{10} - \frac{3}{52} a^{9} - \frac{1}{13} a^{8} - \frac{31}{65} a^{7} - \frac{63}{260} a^{6} + \frac{37}{130} a^{5} + \frac{22}{65} a^{4} - \frac{14}{65} a^{3} - \frac{23}{65} a^{2} + \frac{3}{26} a - \frac{5}{13}$, $\frac{1}{425389640} a^{15} - \frac{67897}{85077928} a^{14} - \frac{2713281}{425389640} a^{13} + \frac{3234843}{212694820} a^{12} + \frac{310463}{212694820} a^{11} + \frac{7419641}{106347410} a^{10} - \frac{411577}{21269482} a^{9} + \frac{982519}{42538964} a^{8} + \frac{27036901}{212694820} a^{7} + \frac{14092517}{42538964} a^{6} + \frac{21854709}{212694820} a^{5} + \frac{17994504}{53173705} a^{4} + \frac{23904376}{53173705} a^{3} - \frac{19032837}{106347410} a^{2} - \frac{2292427}{21269482} a - \frac{37011999}{106347410}$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{92679}{240605} a^{15} - \frac{79319}{481210} a^{14} + \frac{329302}{240605} a^{13} + \frac{470055}{48121} a^{12} - \frac{5585673}{240605} a^{11} + \frac{462146}{240605} a^{10} - \frac{4857677}{240605} a^{9} + \frac{47669207}{481210} a^{8} - \frac{13356072}{240605} a^{7} - \frac{13405983}{240605} a^{6} - \frac{30742}{240605} a^{5} + \frac{14102562}{240605} a^{4} - \frac{794594}{240605} a^{3} - \frac{2631782}{240605} a^{2} - \frac{1315626}{240605} a + \frac{655312}{240605} \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2548.59228871 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_2^2\wr C_2$ (as 16T39):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{10}) \), 4.0.12800.1, 4.0.1280.1, 4.0.320.1, 4.0.512.1, 4.2.25600.1 x2, 4.0.2560.2 x2, \(\Q(i, \sqrt{10})\), 8.0.2621440000.10, 8.0.2621440000.7, 8.0.26214400.1, 8.0.655360000.2, 8.0.104857600.2, 8.0.2621440000.8, 8.0.163840000.2

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

Sibling fields

Galois closure: Deg 32
Degree 8 siblings: 8.0.26214400.2, 8.0.104857600.2, 8.0.26214400.1, 8.0.104857600.1, 8.0.655360000.1, 8.0.655360000.2, 8.0.2621440000.7, 8.0.2621440000.6
Degree 16 siblings: 16.0.429496729600000000.4, 16.4.6871947673600000000.8, 16.0.43980465111040000.1, 16.0.27487790694400000000.3, 16.0.27487790694400000000.2, 16.0.6871947673600000000.13

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$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}$