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Properties

Label 16.16.2553266826...6816.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{64}\cdot 7^{12}$
Root discriminant $68.86$
Ramified primes $2, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2401, 0, -19208, 0, 57624, 0, -79576, 0, 49392, 0, -11368, 0, 1176, 0, -56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 56*x^14 + 1176*x^12 - 11368*x^10 + 49392*x^8 - 79576*x^6 + 57624*x^4 - 19208*x^2 + 2401)
 
gp: K = bnfinit(x^16 - 56*x^14 + 1176*x^12 - 11368*x^10 + 49392*x^8 - 79576*x^6 + 57624*x^4 - 19208*x^2 + 2401, 1)
 

Normalized defining polynomial

\( x^{16} - 56 x^{14} + 1176 x^{12} - 11368 x^{10} + 49392 x^{8} - 79576 x^{6} + 57624 x^{4} - 19208 x^{2} + 2401 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(255326682647558617373989666816=2^{64}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.86$
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:  $2, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7} a^{4}$, $\frac{1}{7} a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{49} a^{8}$, $\frac{1}{49} a^{9}$, $\frac{1}{49} a^{10}$, $\frac{1}{49} a^{11}$, $\frac{1}{343} a^{12}$, $\frac{1}{343} a^{13}$, $\frac{1}{76489} a^{14} - \frac{82}{76489} a^{12} - \frac{69}{10927} a^{10} - \frac{53}{10927} a^{8} + \frac{58}{1561} a^{6} - \frac{10}{1561} a^{4} + \frac{14}{223} a^{2} + \frac{26}{223}$, $\frac{1}{76489} a^{15} - \frac{82}{76489} a^{13} - \frac{69}{10927} a^{11} - \frac{53}{10927} a^{9} + \frac{58}{1561} a^{7} - \frac{10}{1561} a^{5} + \frac{14}{223} a^{3} + \frac{26}{223} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 6311639839.73 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

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 $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{14}) \), \(\Q(\zeta_{16})^+\), 4.4.100352.1, \(\Q(\sqrt{2}, \sqrt{7})\), 8.8.40282095616.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$