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Properties

Label 16.0.102400000000000000.2
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 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 8, -8, 16, -54, 120, -154, 114, -28, -30, 38, -19, 4, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 + 4*x^13 - 19*x^12 + 38*x^11 - 30*x^10 - 28*x^9 + 114*x^8 - 154*x^7 + 120*x^6 - 54*x^5 + 16*x^4 - 8*x^3 + 8*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 + 4*x^13 - 19*x^12 + 38*x^11 - 30*x^10 - 28*x^9 + 114*x^8 - 154*x^7 + 120*x^6 - 54*x^5 + 16*x^4 - 8*x^3 + 8*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} + 4 x^{13} - 19 x^{12} + 38 x^{11} - 30 x^{10} - 28 x^{9} + 114 x^{8} - 154 x^{7} + 120 x^{6} - 54 x^{5} + 16 x^{4} - 8 x^{3} + 8 x^{2} - 4 x + 1 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{704159} a^{15} - \frac{71445}{704159} a^{14} - \frac{203454}{704159} a^{13} + \frac{114048}{704159} a^{12} - \frac{107494}{704159} a^{11} + \frac{135826}{704159} a^{10} + \frac{198231}{704159} a^{9} - \frac{171553}{704159} a^{8} - \frac{330461}{704159} a^{7} + \frac{82117}{704159} a^{6} - \frac{336082}{704159} a^{5} + \frac{292690}{704159} a^{4} + \frac{54010}{704159} a^{3} + \frac{154882}{704159} a^{2} - \frac{80192}{704159} a + \frac{119428}{704159}$

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:  \( -\frac{451291}{704159} a^{15} - \frac{250956}{704159} a^{14} + \frac{258786}{704159} a^{13} - \frac{2558817}{704159} a^{12} + \frac{2969562}{704159} a^{11} - \frac{714575}{704159} a^{10} - \frac{11956769}{704159} a^{9} + \frac{17055166}{704159} a^{8} - \frac{8209367}{704159} a^{7} - \frac{17083011}{704159} a^{6} + \frac{22395463}{704159} a^{5} - \frac{14892432}{704159} a^{4} + \frac{1645193}{704159} a^{3} - \frac{1326163}{704159} a^{2} + \frac{380226}{704159} a - \frac{1155847}{704159} \) (order $20$)
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:  \( 694.921858205 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{5})\), \(\Q(i, \sqrt{5})\), 8.4.320000000.1 x2, 8.0.20000000.1 x2, \(\Q(\zeta_{20})\)

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$