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Properties

Label 16.0.20542695432781824.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{14}$
Root discriminant $10.46$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
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![1, -8, 26, -48, 80, -172, 328, -444, 451, -372, 256, -152, 80, -36, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 36*x^13 + 80*x^12 - 152*x^11 + 256*x^10 - 372*x^9 + 451*x^8 - 444*x^7 + 328*x^6 - 172*x^5 + 80*x^4 - 48*x^3 + 26*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 14*x^14 - 36*x^13 + 80*x^12 - 152*x^11 + 256*x^10 - 372*x^9 + 451*x^8 - 444*x^7 + 328*x^6 - 172*x^5 + 80*x^4 - 48*x^3 + 26*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 14 x^{14} - 36 x^{13} + 80 x^{12} - 152 x^{11} + 256 x^{10} - 372 x^{9} + 451 x^{8} - 444 x^{7} + 328 x^{6} - 172 x^{5} + 80 x^{4} - 48 x^{3} + 26 x^{2} - 8 x + 1 \)

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:  \(20542695432781824=2^{32}\cdot 3^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.46$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{3}{11} a^{13} - \frac{2}{11} a^{12} + \frac{1}{11} a^{11} - \frac{3}{11} a^{10} - \frac{3}{11} a^{9} - \frac{5}{11} a^{8} + \frac{3}{11} a^{7} + \frac{2}{11} a^{6} + \frac{3}{11} a^{5} - \frac{3}{11} a^{4} - \frac{5}{11} a^{3} + \frac{4}{11} a^{2} - \frac{1}{11} a - \frac{5}{11}$, $\frac{1}{4830529} a^{15} - \frac{199241}{4830529} a^{14} + \frac{1965782}{4830529} a^{13} - \frac{1608440}{4830529} a^{12} + \frac{2149083}{4830529} a^{11} - \frac{515819}{4830529} a^{10} + \frac{1165023}{4830529} a^{9} + \frac{120984}{439139} a^{8} + \frac{2308224}{4830529} a^{7} + \frac{1618245}{4830529} a^{6} + \frac{30295}{210023} a^{5} - \frac{1219425}{4830529} a^{4} - \frac{1268640}{4830529} a^{3} + \frac{1719429}{4830529} a^{2} + \frac{214809}{4830529} a + \frac{2342755}{4830529}$

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:  \( \frac{116163748}{4830529} a^{15} - \frac{400145791}{4830529} a^{14} + \frac{1400761486}{4830529} a^{13} - \frac{3394148708}{4830529} a^{12} + \frac{7372792822}{4830529} a^{11} - \frac{13482010844}{4830529} a^{10} + \frac{22077378896}{4830529} a^{9} - \frac{30647044013}{4830529} a^{8} + \frac{34879441088}{4830529} a^{7} - \frac{31554781848}{4830529} a^{6} + \frac{864222976}{210023} a^{5} - \frac{8354939592}{4830529} a^{4} + \frac{4347436208}{4830529} a^{3} - \frac{3058729513}{4830529} a^{2} + \frac{1248592326}{4830529} a - \frac{189141914}{4830529} \) (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:  \( 160.565615792 \)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 4.2.1728.1 x2, 4.0.432.1 x2, 8.0.2985984.1, 8.2.143327232.1 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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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