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Properties

Label 16.0.102400000000000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{14}$
Root discriminant $11.56$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 24, -76, 191, -378, 596, -738, 697, -482, 226, -52, -9, 6, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 6*x^13 - 9*x^12 - 52*x^11 + 226*x^10 - 482*x^9 + 697*x^8 - 738*x^7 + 596*x^6 - 378*x^5 + 191*x^4 - 76*x^3 + 24*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 + 6*x^13 - 9*x^12 - 52*x^11 + 226*x^10 - 482*x^9 + 697*x^8 - 738*x^7 + 596*x^6 - 378*x^5 + 191*x^4 - 76*x^3 + 24*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} + 6 x^{13} - 9 x^{12} - 52 x^{11} + 226 x^{10} - 482 x^{9} + 697 x^{8} - 738 x^{7} + 596 x^{6} - 378 x^{5} + 191 x^{4} - 76 x^{3} + 24 x^{2} - 6 x + 1 \)

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:  \(102400000000000000=2^{24}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.56$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} + \frac{2}{11} a^{13} - \frac{5}{11} a^{12} - \frac{3}{11} a^{10} - \frac{4}{11} a^{9} + \frac{1}{11} a^{8} + \frac{4}{11} a^{7} - \frac{4}{11} a^{6} + \frac{1}{11} a^{5} + \frac{4}{11} a^{4} - \frac{1}{11} a^{3} + \frac{2}{11} a^{2} + \frac{1}{11} a - \frac{1}{11}$, $\frac{1}{4374469} a^{15} - \frac{99708}{4374469} a^{14} + \frac{1299831}{4374469} a^{13} + \frac{1656971}{4374469} a^{12} + \frac{1150377}{4374469} a^{11} + \frac{1388720}{4374469} a^{10} - \frac{1041224}{4374469} a^{9} - \frac{303415}{4374469} a^{8} + \frac{1043685}{4374469} a^{7} - \frac{99085}{4374469} a^{6} + \frac{2018113}{4374469} a^{5} + \frac{1693095}{4374469} a^{4} + \frac{2017342}{4374469} a^{3} - \frac{196696}{397679} a^{2} + \frac{207508}{4374469} a + \frac{667695}{4374469}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{6863158}{4374469} a^{15} - \frac{20730416}{4374469} a^{14} + \frac{6586650}{4374469} a^{13} + \frac{49348837}{4374469} a^{12} - \frac{15017270}{4374469} a^{11} - \frac{373017604}{4374469} a^{10} + \frac{1187481022}{4374469} a^{9} - \frac{2122258780}{4374469} a^{8} + \frac{2605590630}{4374469} a^{7} - \frac{2279644310}{4374469} a^{6} + \frac{1489815532}{4374469} a^{5} - \frac{713301184}{4374469} a^{4} + \frac{245897576}{4374469} a^{3} - \frac{51073010}{4374469} a^{2} + \frac{9653340}{4374469} a + \frac{2191268}{4374469} \) (order $10$)
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:  \( 309.712108606 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.2.400.1, 4.2.2000.1, 8.0.20000000.1 x2, 8.0.4000000.2

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed