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Properties

Label 16.0.7771775074107392.1
Degree $16$
Signature $[0, 8]$
Discriminant $7.772\times 10^{15}$
Root discriminant $9.84$
Ramified primes $2, 17$
Class number $1$
Class group trivial
Galois group $C_4.D_4:C_4$ (as 16T260)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 17*x^14 - 28*x^13 + 26*x^12 - 10*x^11 - 3*x^10 + 2*x^9 + x^8 + 6*x^7 - 10*x^6 + 2*x^5 + 6*x^4 - 4*x^3 - 2*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 17*x^14 - 28*x^13 + 26*x^12 - 10*x^11 - 3*x^10 + 2*x^9 + x^8 + 6*x^7 - 10*x^6 + 2*x^5 + 6*x^4 - 4*x^3 - 2*x^2 + 2*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -2, -4, 6, 2, -10, 6, 1, 2, -3, -10, 26, -28, 17, -6, 1]);
 

\( x^{16} - 6 x^{15} + 17 x^{14} - 28 x^{13} + 26 x^{12} - 10 x^{11} - 3 x^{10} + 2 x^{9} + x^{8} + 6 x^{7} - 10 x^{6} + 2 x^{5} + 6 x^{4} - 4 x^{3} - 2 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(7771775074107392\)\(\medspace = 2^{16}\cdot 17^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $9.84$
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:  $2, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{67} a^{15} - \frac{29}{67} a^{14} + \frac{14}{67} a^{13} - \frac{15}{67} a^{12} - \frac{31}{67} a^{11} + \frac{33}{67} a^{10} - \frac{25}{67} a^{9} - \frac{26}{67} a^{8} - \frac{4}{67} a^{7} + \frac{31}{67} a^{6} + \frac{14}{67} a^{5} + \frac{15}{67} a^{4} - \frac{4}{67} a^{3} + \frac{21}{67} a^{2} - \frac{16}{67} a - \frac{32}{67}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{44}{67} a^{15} - \frac{137}{67} a^{14} + \frac{80}{67} a^{13} + \frac{546}{67} a^{12} - \frac{1699}{67} a^{11} + \frac{2390}{67} a^{10} - \frac{1971}{67} a^{9} + \frac{1067}{67} a^{8} - \frac{779}{67} a^{7} + \frac{694}{67} a^{6} - \frac{54}{67} a^{5} - \frac{546}{67} a^{4} + \frac{427}{67} a^{3} + \frac{53}{67} a^{2} - \frac{168}{67} a - \frac{68}{67} \) (order $4$)
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 25.2889722607 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 25.2889722607 \cdot 1}{4\sqrt{7771775074107392}}\approx 0.174200751443$

Galois group

$C_4.D_4:C_4$ (as 16T260):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.272.1, 8.0.1257728.1

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

Sibling fields

Degree 16 sibling: 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 $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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

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
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.7.4$x^{8} - 12393$$8$$1$$7$$C_8$$[\ ]_{8}$