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Properties

Label 16.0.6158959248447369.2
Degree $16$
Signature $[0, 8]$
Discriminant $6.159\times 10^{15}$
Root discriminant $9.70$
Ramified primes $3, 7, 13$
Class number $1$
Class group trivial
Galois group $C_4.C_2^2:D_4$ (as 16T211)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 17*x^12 - 18*x^11 + 13*x^10 + 3*x^9 - 29*x^8 + 45*x^7 - 34*x^6 + 3*x^5 + 23*x^4 - 27*x^3 + 17*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 17*x^12 - 18*x^11 + 13*x^10 + 3*x^9 - 29*x^8 + 45*x^7 - 34*x^6 + 3*x^5 + 23*x^4 - 27*x^3 + 17*x^2 - 6*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 17, -27, 23, 3, -34, 45, -29, 3, 13, -18, 17, -12, 7, -3, 1]);
 

\( x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 17 x^{12} - 18 x^{11} + 13 x^{10} + 3 x^{9} - 29 x^{8} + 45 x^{7} - 34 x^{6} + 3 x^{5} + 23 x^{4} - 27 x^{3} + 17 x^{2} - 6 x + 1 \)

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:  \(6158959248447369\)\(\medspace = 3^{12}\cdot 7^{4}\cdot 13^{6}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $9.70$
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:  $3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{258287} a^{15} + \frac{30004}{258287} a^{14} - \frac{58447}{258287} a^{13} - \frac{50411}{258287} a^{12} + \frac{104099}{258287} a^{11} - \frac{24303}{258287} a^{10} - \frac{115907}{258287} a^{9} + \frac{71396}{258287} a^{8} - \frac{110922}{258287} a^{7} + \frac{108160}{258287} a^{6} - \frac{77356}{258287} a^{5} + \frac{3780}{258287} a^{4} + \frac{38490}{258287} a^{3} - \frac{90061}{258287} a^{2} - \frac{3529}{258287} a + \frac{2961}{258287}$

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:  \( -\frac{2168}{1427} a^{15} + \frac{6831}{1427} a^{14} - \frac{15920}{1427} a^{13} + \frac{27085}{1427} a^{12} - \frac{37976}{1427} a^{11} + \frac{39739}{1427} a^{10} - \frac{26875}{1427} a^{9} - \frac{9827}{1427} a^{8} + \frac{69352}{1427} a^{7} - \frac{106130}{1427} a^{6} + \frac{76691}{1427} a^{5} + \frac{4502}{1427} a^{4} - \frac{63856}{1427} a^{3} + \frac{62907}{1427} a^{2} - \frac{32096}{1427} a + \frac{7760}{1427} \) (order $6$)
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:  \( 33.0234170129 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 33.0234170129 \cdot 1}{6\sqrt{6158959248447369}}\approx 0.170355453107$

Galois group

$C_4.C_2^2:D_4$ (as 16T211):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.117.1, 4.0.189.1, 4.0.2457.1, 8.0.1601613.1, 8.0.8719893.1, 8.0.6036849.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3Data not computed
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$