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Properties

Label 16.0.45263227346...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $14.65$
Ramified primes $3, 5, 41$
Class number $1$
Class group Trivial
Galois group $C_2 \times (C_2^2:C_4)$ (as 16T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -384, 448, -192, 48, 48, 56, -96, 113, -48, 14, 6, 3, -6, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 - 6*x^13 + 3*x^12 + 6*x^11 + 14*x^10 - 48*x^9 + 113*x^8 - 96*x^7 + 56*x^6 + 48*x^5 + 48*x^4 - 192*x^3 + 448*x^2 - 384*x + 256)
 
gp: K = bnfinit(x^16 - 3*x^15 + 7*x^14 - 6*x^13 + 3*x^12 + 6*x^11 + 14*x^10 - 48*x^9 + 113*x^8 - 96*x^7 + 56*x^6 + 48*x^5 + 48*x^4 - 192*x^3 + 448*x^2 - 384*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 7 x^{14} - 6 x^{13} + 3 x^{12} + 6 x^{11} + 14 x^{10} - 48 x^{9} + 113 x^{8} - 96 x^{7} + 56 x^{6} + 48 x^{5} + 48 x^{4} - 192 x^{3} + 448 x^{2} - 384 x + 256 \)

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:  \(4526322734619140625=3^{8}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.65$
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:  $3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{10} - \frac{1}{16} a^{8} + \frac{1}{4} a^{7} - \frac{3}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} + \frac{1}{32} a^{11} - \frac{1}{32} a^{9} + \frac{1}{8} a^{8} - \frac{3}{16} a^{7} - \frac{3}{8} a^{6} - \frac{15}{32} a^{5} - \frac{7}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{13} + \frac{1}{64} a^{12} - \frac{1}{64} a^{10} + \frac{1}{16} a^{9} + \frac{13}{32} a^{8} - \frac{3}{16} a^{7} - \frac{15}{64} a^{6} - \frac{7}{32} a^{5} + \frac{3}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{30592} a^{15} + \frac{137}{30592} a^{14} + \frac{67}{30592} a^{13} - \frac{93}{15296} a^{12} - \frac{1181}{30592} a^{11} - \frac{1407}{15296} a^{10} - \frac{37}{15296} a^{9} + \frac{83}{239} a^{8} - \frac{11535}{30592} a^{7} - \frac{3185}{7648} a^{6} - \frac{673}{3824} a^{5} - \frac{1243}{3824} a^{4} + \frac{57}{478} a^{3} - \frac{59}{956} a^{2} - \frac{30}{239} a + \frac{99}{239}$

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:  \( -\frac{1045}{30592} a^{15} + \frac{2147}{30592} a^{14} - \frac{3095}{30592} a^{13} - \frac{1283}{15296} a^{12} + \frac{8553}{30592} a^{11} - \frac{5749}{15296} a^{10} - \frac{6267}{15296} a^{9} + \frac{1283}{956} a^{8} - \frac{56533}{30592} a^{7} - \frac{467}{7648} a^{6} + \frac{4495}{1912} a^{5} - \frac{12219}{3824} a^{4} - \frac{825}{956} a^{3} + \frac{2745}{478} a^{2} - \frac{1871}{239} a + \frac{988}{239} \) (order $30$)
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:  \( 5126.68171823 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:C_4$ (as 16T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_2^2:C_4)$
Character table for $C_2 \times (C_2^2:C_4)$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{5})\), 4.0.1025.1, 4.4.9225.1, \(\Q(\zeta_{15})^+\), 4.4.5125.1, 4.0.46125.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.2127515625.3, 8.0.26265625.1, 8.8.2127515625.1, 8.0.2127515625.1, 8.0.2127515625.2, \(\Q(\zeta_{15})\), 8.0.85100625.1

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$