Properties

Label 16.16.321...808.1
Degree $16$
Signature $[16, 0]$
Discriminant $3.212\times 10^{29}$
Root discriminant $69.85$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^14 + 936*x^12 - 9504*x^10 + 53604*x^8 - 166752*x^6 + 270864*x^4 - 202176*x^2 + 46818)
 
gp: K = bnfinit(x^16 - 48*x^14 + 936*x^12 - 9504*x^10 + 53604*x^8 - 166752*x^6 + 270864*x^4 - 202176*x^2 + 46818, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![46818, 0, -202176, 0, 270864, 0, -166752, 0, 53604, 0, -9504, 0, 936, 0, -48, 0, 1]);
 

\( x^{16} - 48 x^{14} + 936 x^{12} - 9504 x^{10} + 53604 x^{8} - 166752 x^{6} + 270864 x^{4} - 202176 x^{2} + 46818 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[16, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(321236373250909071617512439808\)\(\medspace = 2^{79}\cdot 3^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $69.85$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4}$, $\frac{1}{3} a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{63} a^{8} - \frac{1}{21} a^{6} - \frac{1}{7} a^{4} + \frac{1}{7} a^{2} - \frac{2}{7}$, $\frac{1}{63} a^{9} - \frac{1}{21} a^{7} - \frac{1}{7} a^{5} + \frac{1}{7} a^{3} - \frac{2}{7} a$, $\frac{1}{63} a^{10} + \frac{1}{21} a^{6} + \frac{1}{21} a^{4} + \frac{1}{7} a^{2} + \frac{1}{7}$, $\frac{1}{63} a^{11} + \frac{1}{21} a^{7} + \frac{1}{21} a^{5} + \frac{1}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{189} a^{12} - \frac{1}{21} a^{6} - \frac{1}{7} a^{4} - \frac{3}{7} a^{2} + \frac{2}{7}$, $\frac{1}{3213} a^{13} + \frac{1}{357} a^{11} + \frac{1}{153} a^{9} - \frac{5}{357} a^{7} - \frac{1}{51} a^{5} + \frac{7}{17} a^{3} - \frac{44}{119} a$, $\frac{1}{3213} a^{14} - \frac{8}{3213} a^{12} + \frac{1}{153} a^{10} + \frac{2}{1071} a^{8} - \frac{1}{51} a^{6} + \frac{4}{51} a^{4} + \frac{24}{119} a^{2} + \frac{3}{7}$, $\frac{1}{3213} a^{15} - \frac{1}{357} a^{11} + \frac{1}{153} a^{9} - \frac{10}{119} a^{7} - \frac{4}{51} a^{5} - \frac{26}{119} a^{3} + \frac{5}{119} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 8601040814.68 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{16}\cdot(2\pi)^{0}\cdot 8601040814.68 \cdot 1}{2\sqrt{321236373250909071617512439808}}\approx 0.497265794585$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.173946175488.1

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

Sibling fields

Galois closure: Deg 32

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 $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ $16$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
3Data not computed