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Properties

Label 16.0.18014398509...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{58}\cdot 5^{4}$
Root discriminant $18.45$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $SD_{16}:C_2$ (as 16T50)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 0, 0, 0, 300, 0, 0, 0, 114, 0, 0, 0, -12, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^12 + 114*x^8 + 300*x^4 + 625)
 
gp: K = bnfinit(x^16 - 12*x^12 + 114*x^8 + 300*x^4 + 625, 1)
 

Normalized defining polynomial

\( x^{16} - 12 x^{12} + 114 x^{8} + 300 x^{4} + 625 \)

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:  \(180143985094819840000=2^{58}\cdot 5^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.45$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{16} a^{8} + \frac{1}{8} a^{4} - \frac{1}{16}$, $\frac{1}{16} a^{9} + \frac{1}{8} a^{5} - \frac{1}{16} a$, $\frac{1}{80} a^{10} + \frac{9}{40} a^{6} - \frac{1}{80} a^{2}$, $\frac{1}{80} a^{11} + \frac{9}{40} a^{7} - \frac{1}{80} a^{3}$, $\frac{1}{6800} a^{12} + \frac{69}{3400} a^{8} - \frac{2561}{6800} a^{4} - \frac{11}{34}$, $\frac{1}{13600} a^{13} - \frac{1}{13600} a^{12} - \frac{287}{13600} a^{9} + \frac{287}{13600} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{3389}{13600} a^{5} - \frac{3389}{13600} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{201}{544} a - \frac{201}{544}$, $\frac{1}{68000} a^{14} - \frac{1}{13600} a^{12} - \frac{287}{68000} a^{10} + \frac{287}{13600} a^{8} + \frac{23789}{68000} a^{6} + \frac{3411}{13600} a^{4} + \frac{473}{2720} a^{2} + \frac{71}{544}$, $\frac{1}{68000} a^{15} - \frac{1}{13600} a^{12} - \frac{287}{68000} a^{11} + \frac{287}{13600} a^{8} - \frac{10211}{68000} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{3389}{13600} a^{4} - \frac{887}{2720} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{201}{544}$

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:  \( -\frac{23}{6800} a^{12} + \frac{113}{3400} a^{8} - \frac{2297}{6800} a^{4} - \frac{18}{17} \) (order $8$)
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:  \( 11125.0816339 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}:C_2$ (as 16T50):

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 $SD_{16}:C_2$
Character table for $SD_{16}:C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.0.512.1 x2, 4.2.1024.1 x2, \(\Q(\zeta_{8})\), 8.0.4194304.1

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 sibling: 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$