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Properties

Label 16.0.13458885161...8464.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{58}\cdot 3^{4}\cdot 7^{8}$
Root discriminant $42.96$
Ramified primes $2, 3, 7$
Class number $72$ (GRH)
Class group $[2, 6, 6]$ (GRH)
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![81, 0, 12312, 0, 36288, 0, 36792, 0, 17782, 0, 4520, 0, 608, 0, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 40*x^14 + 608*x^12 + 4520*x^10 + 17782*x^8 + 36792*x^6 + 36288*x^4 + 12312*x^2 + 81)
 
gp: K = bnfinit(x^16 + 40*x^14 + 608*x^12 + 4520*x^10 + 17782*x^8 + 36792*x^6 + 36288*x^4 + 12312*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{16} + 40 x^{14} + 608 x^{12} + 4520 x^{10} + 17782 x^{8} + 36792 x^{6} + 36288 x^{4} + 12312 x^{2} + 81 \)

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:  \(134588851614250885095358464=2^{58}\cdot 3^{4}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.96$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{6} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{12} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{36} a^{12} - \frac{1}{18} a^{10} - \frac{1}{36} a^{8} + \frac{1}{18} a^{6} + \frac{1}{36} a^{4} - \frac{1}{4}$, $\frac{1}{36} a^{13} + \frac{1}{36} a^{11} - \frac{1}{12} a^{10} + \frac{1}{18} a^{9} - \frac{1}{12} a^{8} + \frac{1}{18} a^{7} + \frac{13}{36} a^{5} - \frac{1}{3} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{19968876} a^{14} - \frac{82793}{19968876} a^{12} - \frac{1315729}{19968876} a^{10} - \frac{1012285}{19968876} a^{8} + \frac{1310863}{19968876} a^{6} + \frac{471103}{6656292} a^{4} + \frac{622637}{2218764} a^{2} - \frac{276485}{739588}$, $\frac{1}{19968876} a^{15} - \frac{82793}{19968876} a^{13} + \frac{87086}{4992219} a^{11} - \frac{1}{12} a^{10} + \frac{162947}{4992219} a^{9} - \frac{1}{12} a^{8} + \frac{1310863}{19968876} a^{7} + \frac{2689867}{6656292} a^{5} - \frac{1}{3} a^{4} - \frac{260359}{554691} a^{3} - \frac{1}{4} a^{2} - \frac{22897}{184897} a - \frac{1}{4}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}$, which has order $72$ (assuming GRH)

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:  \( -1 \) (order $2$)
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 44651.6325002 \) (assuming GRH)
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

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{14}) \), 4.4.25088.1 x2, 4.4.7168.1 x2, \(\Q(\sqrt{2}, \sqrt{7})\), 8.8.10070523904.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{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}$