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Properties

Label 16.0.13785969539...7936.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{58}\cdot 3^{14}$
Root discriminant $32.26$
Ramified primes $2, 3$
Class number $18$ (GRH)
Class group $[3, 6]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81796, 0, -59584, 0, 37384, 0, -10576, 0, 3136, 0, -304, 0, 76, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 + 76*x^12 - 304*x^10 + 3136*x^8 - 10576*x^6 + 37384*x^4 - 59584*x^2 + 81796)
 
gp: K = bnfinit(x^16 + 8*x^14 + 76*x^12 - 304*x^10 + 3136*x^8 - 10576*x^6 + 37384*x^4 - 59584*x^2 + 81796, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} + 76 x^{12} - 304 x^{10} + 3136 x^{8} - 10576 x^{6} + 37384 x^{4} - 59584 x^{2} + 81796 \)

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:  \(1378596953991976568487936=2^{58}\cdot 3^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.26$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{16} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8}$, $\frac{1}{16} a^{9} + \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{6} - \frac{1}{2} a^{4} - \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{32} a^{11} - \frac{1}{16} a^{7} - \frac{1}{4} a^{5} - \frac{1}{16} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{12} - \frac{1}{2} a^{7} - \frac{3}{16} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{8}$, $\frac{1}{32} a^{13} - \frac{3}{16} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{8} a$, $\frac{1}{1345848914272} a^{14} - \frac{510716189}{84115557142} a^{12} + \frac{2972779701}{168231114284} a^{10} - \frac{11588505247}{672924457136} a^{8} - \frac{1}{2} a^{7} - \frac{199291602375}{672924457136} a^{6} - \frac{1}{2} a^{5} - \frac{18247334071}{336462228568} a^{4} + \frac{155620980459}{336462228568} a^{2} + \frac{116857452015}{336462228568}$, $\frac{1}{192456394740896} a^{15} + \frac{2305006362381}{192456394740896} a^{13} + \frac{48003337973}{48114098685224} a^{11} - \frac{768628519525}{96228197370448} a^{9} + \frac{34288086825845}{96228197370448} a^{7} - \frac{1}{2} a^{6} - \frac{11518268218025}{96228197370448} a^{5} - \frac{1}{2} a^{4} - \frac{4723081333777}{48114098685224} a^{3} - \frac{2451030087223}{6014262335653} a$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( 20146.2577766 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.13824.1 x2, 4.4.27648.1 x2, 8.8.3057647616.1, 8.0.293534171136.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed