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Properties

Label 16.0.231549328645449984.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 3^{8}\cdot 13^{10}$
Root discriminant $12.17$
Ramified primes $2, 3, 13$
Class number $1$
Class group Trivial
Galois group $OD_{16}:C_2^2$ (as 16T106)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, 26, 49, 63, 45, 4, -9, -18, -23, -2, -15, 10, -6, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 6*x^14 - 6*x^13 + 10*x^12 - 15*x^11 - 2*x^10 - 23*x^9 - 18*x^8 - 9*x^7 + 4*x^6 + 45*x^5 + 63*x^4 + 49*x^3 + 26*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^16 - x^15 + 6*x^14 - 6*x^13 + 10*x^12 - 15*x^11 - 2*x^10 - 23*x^9 - 18*x^8 - 9*x^7 + 4*x^6 + 45*x^5 + 63*x^4 + 49*x^3 + 26*x^2 + 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 6 x^{14} - 6 x^{13} + 10 x^{12} - 15 x^{11} - 2 x^{10} - 23 x^{9} - 18 x^{8} - 9 x^{7} + 4 x^{6} + 45 x^{5} + 63 x^{4} + 49 x^{3} + 26 x^{2} + 8 x + 1 \)

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:  \(231549328645449984=2^{8}\cdot 3^{8}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.17$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{13} - \frac{3}{7} a^{12} - \frac{2}{7} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{9} + \frac{3}{7} a^{8} - \frac{2}{7} a^{6} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{888881} a^{15} - \frac{12315}{888881} a^{14} - \frac{224752}{888881} a^{13} + \frac{255603}{888881} a^{12} - \frac{348660}{888881} a^{11} - \frac{149971}{888881} a^{10} + \frac{29123}{888881} a^{9} + \frac{15190}{126983} a^{8} + \frac{355024}{888881} a^{7} - \frac{35541}{126983} a^{6} + \frac{225230}{888881} a^{5} - \frac{173455}{888881} a^{4} - \frac{310076}{888881} a^{3} + \frac{278052}{888881} a^{2} + \frac{418259}{888881} a + \frac{52873}{126983}$

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:  \( -\frac{1946981}{126983} a^{15} + \frac{2680615}{126983} a^{14} - \frac{12686993}{126983} a^{13} + \frac{16446210}{126983} a^{12} - \frac{25611350}{126983} a^{11} + \frac{38738965}{126983} a^{10} - \frac{10494296}{126983} a^{9} + \frac{48434404}{126983} a^{8} + \frac{17136596}{126983} a^{7} + \frac{10718037}{126983} a^{6} - \frac{11565271}{126983} a^{5} - \frac{83412670}{126983} a^{4} - \frac{91195563}{126983} a^{3} - \frac{60987323}{126983} a^{2} - \frac{27775407}{126983} a - \frac{5099267}{126983} \) (order $6$)
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:  \( 223.591496429 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}:C_2^2$ (as 16T106):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $OD_{16}:C_2^2$
Character table for $OD_{16}:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-39}) \), 4.2.507.1 x2, 4.0.117.1 x2, \(\Q(\sqrt{-3}, \sqrt{13})\), 8.0.2313441.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$