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Properties

Label 16.0.45514779957...3536.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{26}\cdot 7^{14}$
Root discriminant $16.93$
Ramified primes $2, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times D_8):C_2$ (as 16T126)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -24, 122, -364, 763, -1246, 1666, -1886, 1842, -1548, 1106, -658, 322, -126, 38, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 38*x^14 - 126*x^13 + 322*x^12 - 658*x^11 + 1106*x^10 - 1548*x^9 + 1842*x^8 - 1886*x^7 + 1666*x^6 - 1246*x^5 + 763*x^4 - 364*x^3 + 122*x^2 - 24*x + 2)
 
gp: K = bnfinit(x^16 - 8*x^15 + 38*x^14 - 126*x^13 + 322*x^12 - 658*x^11 + 1106*x^10 - 1548*x^9 + 1842*x^8 - 1886*x^7 + 1666*x^6 - 1246*x^5 + 763*x^4 - 364*x^3 + 122*x^2 - 24*x + 2, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 38 x^{14} - 126 x^{13} + 322 x^{12} - 658 x^{11} + 1106 x^{10} - 1548 x^{9} + 1842 x^{8} - 1886 x^{7} + 1666 x^{6} - 1246 x^{5} + 763 x^{4} - 364 x^{3} + 122 x^{2} - 24 x + 2 \)

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:  \(45514779957485633536=2^{26}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.93$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} + \frac{2}{5} a^{11} + \frac{1}{10} a^{10} - \frac{2}{5} a^{8} - \frac{1}{10} a^{6} + \frac{2}{5} a^{5} + \frac{3}{10} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5}$, $\frac{1}{10} a^{13} - \frac{1}{2} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{6} - \frac{3}{10} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{13} + \frac{1}{20} a^{11} - \frac{1}{4} a^{10} + \frac{3}{20} a^{8} + \frac{9}{20} a^{7} + \frac{1}{5} a^{6} + \frac{7}{20} a^{5} - \frac{7}{20} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{3}{10}$, $\frac{1}{1060} a^{15} + \frac{19}{1060} a^{14} - \frac{8}{265} a^{13} + \frac{17}{1060} a^{12} + \frac{39}{1060} a^{11} + \frac{6}{265} a^{10} + \frac{111}{1060} a^{9} - \frac{247}{1060} a^{8} - \frac{1}{265} a^{7} - \frac{49}{212} a^{6} + \frac{33}{1060} a^{5} - \frac{49}{265} a^{4} + \frac{41}{530} a^{3} + \frac{77}{530} a^{2} - \frac{139}{530} a - \frac{54}{265}$

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:  $7$
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:  \( 6826.444479 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_8):C_2$ (as 16T126):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $(C_2\times D_8):C_2$
Character table for $(C_2\times D_8):C_2$

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.2744.1, 4.0.1372.1, 4.0.392.1, 8.0.1686616064.2, 8.0.6746464256.1, 8.0.30118144.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
7Data not computed