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Properties

Label 16.0.96826519964...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}$
Root discriminant $20.49$
Ramified primes $2, 3, 5, 7$
Class number $4$
Class group $[4]$
Galois group $Q_8 : C_2^2$ (as 16T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, 0, 0, 1896, 0, 0, 0, 737, 0, 0, 0, 54, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 54*x^12 + 737*x^8 + 1896*x^4 + 16)
 
gp: K = bnfinit(x^16 + 54*x^12 + 737*x^8 + 1896*x^4 + 16, 1)
 

Normalized defining polynomial

\( x^{16} + 54 x^{12} + 737 x^{8} + 1896 x^{4} + 16 \)

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:  \(968265199641600000000=2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.49$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{7} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{3}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{16} a^{7} - \frac{3}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{15808} a^{12} + \frac{333}{7904} a^{8} - \frac{1}{4} a^{5} - \frac{37}{832} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1863}{3952}$, $\frac{1}{15808} a^{13} + \frac{333}{7904} a^{9} - \frac{37}{832} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{113}{3952} a - \frac{1}{2}$, $\frac{1}{63232} a^{14} - \frac{1}{31616} a^{13} - \frac{1}{31616} a^{12} - \frac{655}{31616} a^{10} + \frac{655}{15808} a^{9} - \frac{333}{15808} a^{8} - \frac{141}{3328} a^{6} + \frac{141}{1664} a^{5} + \frac{245}{1664} a^{4} - \frac{5815}{15808} a^{2} + \frac{1863}{7904} a - \frac{113}{7904}$, $\frac{1}{126464} a^{15} - \frac{1}{63232} a^{13} - \frac{1}{31616} a^{12} + \frac{1321}{63232} a^{11} - \frac{1321}{31616} a^{9} + \frac{655}{15808} a^{8} - \frac{349}{6656} a^{7} - \frac{1}{8} a^{6} + \frac{349}{3328} a^{5} + \frac{141}{1664} a^{4} + \frac{6041}{31616} a^{3} - \frac{1}{8} a^{2} + \frac{5815}{15808} a + \frac{3839}{7904}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( \frac{21}{7904} a^{14} - \frac{1}{1216} a^{12} + \frac{571}{3952} a^{10} - \frac{29}{608} a^{8} + \frac{835}{416} a^{6} - \frac{43}{64} a^{4} + \frac{10771}{1976} a^{2} - \frac{265}{304} \) (order $12$)
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:  \( 31173.385849 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2$ (as 16T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 17 conjugacy class representatives for $Q_8 : C_2^2$
Character table for $Q_8 : C_2^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{-3}, \sqrt{35})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(i, \sqrt{35})\), \(\Q(\sqrt{3}, \sqrt{35})\), \(\Q(i, \sqrt{105})\), \(\Q(\sqrt{3}, \sqrt{-35})\), 8.0.31116960000.10, 8.0.25401600.1 x2, 8.4.31116960000.1 x2

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 R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$