Properties

Label 16.0.4096000000000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}$
Root discriminant $9.46$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 18, -36, 47, -30, -18, 72, -95, 72, -18, -30, 47, -36, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 47*x^12 - 30*x^11 - 18*x^10 + 72*x^9 - 95*x^8 + 72*x^7 - 18*x^6 - 30*x^5 + 47*x^4 - 36*x^3 + 18*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 47*x^12 - 30*x^11 - 18*x^10 + 72*x^9 - 95*x^8 + 72*x^7 - 18*x^6 - 30*x^5 + 47*x^4 - 36*x^3 + 18*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 47 x^{12} - 30 x^{11} - 18 x^{10} + 72 x^{9} - 95 x^{8} + 72 x^{7} - 18 x^{6} - 30 x^{5} + 47 x^{4} - 36 x^{3} + 18 x^{2} - 6 x + 1 \)

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:  \(4096000000000000=2^{24}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $9.46$
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]~
 
$|\Gal(K/\Q)|$:  $16$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{101} a^{14} + \frac{30}{101} a^{13} - \frac{14}{101} a^{12} + \frac{36}{101} a^{11} + \frac{44}{101} a^{10} + \frac{3}{101} a^{9} + \frac{46}{101} a^{8} + \frac{8}{101} a^{7} + \frac{46}{101} a^{6} + \frac{3}{101} a^{5} + \frac{44}{101} a^{4} + \frac{36}{101} a^{3} - \frac{14}{101} a^{2} + \frac{30}{101} a + \frac{1}{101}$, $\frac{1}{101} a^{15} - \frac{5}{101} a^{13} - \frac{49}{101} a^{12} - \frac{26}{101} a^{11} - \frac{4}{101} a^{10} - \frac{44}{101} a^{9} + \frac{42}{101} a^{8} + \frac{8}{101} a^{7} + \frac{37}{101} a^{6} - \frac{46}{101} a^{5} + \frac{29}{101} a^{4} + \frac{17}{101} a^{3} + \frac{46}{101} a^{2} + \frac{10}{101} a - \frac{30}{101}$

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:  \( -\frac{174}{101} a^{15} + \frac{807}{101} a^{14} - \frac{1887}{101} a^{13} + \frac{2985}{101} a^{12} - \frac{2380}{101} a^{11} - \frac{863}{101} a^{10} + \frac{4421}{101} a^{9} - \frac{5940}{101} a^{8} + \frac{3953}{101} a^{7} - \frac{424}{101} a^{6} - \frac{2705}{101} a^{5} + \frac{2889}{101} a^{4} - \frac{1580}{101} a^{3} + \frac{292}{101} a^{2} - \frac{53}{101} a - \frac{33}{101} \) (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:  \( 87.2940978199 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

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 $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{5})\), 4.2.400.1 x2, 4.0.320.1 x2, 4.0.8000.1 x2, 4.2.2000.1 x2, \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{5})\), 8.0.2560000.1, 8.0.64000000.3, \(\Q(\zeta_{20})\), 8.4.64000000.1 x2, 8.0.4000000.2 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$