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Properties

Label 16.0.89690541287...6496.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 13^{8}$
Root discriminant $20.40$
Ramified primes $2, 13$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_8:C_2^2$ (as 16T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![243, -432, 36, 732, -604, -288, 648, -176, -131, 32, 40, -24, 8, -8, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 8*x^13 + 8*x^12 - 24*x^11 + 40*x^10 + 32*x^9 - 131*x^8 - 176*x^7 + 648*x^6 - 288*x^5 - 604*x^4 + 732*x^3 + 36*x^2 - 432*x + 243)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 8*x^13 + 8*x^12 - 24*x^11 + 40*x^10 + 32*x^9 - 131*x^8 - 176*x^7 + 648*x^6 - 288*x^5 - 604*x^4 + 732*x^3 + 36*x^2 - 432*x + 243, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{14} - 8 x^{13} + 8 x^{12} - 24 x^{11} + 40 x^{10} + 32 x^{9} - 131 x^{8} - 176 x^{7} + 648 x^{6} - 288 x^{5} - 604 x^{4} + 732 x^{3} + 36 x^{2} - 432 x + 243 \)

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:  \(896905412873600106496=2^{40}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.40$
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:  $2, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{4}{9} a^{6} - \frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{2}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{2}{9} a^{3} + \frac{4}{9} a^{2}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{4}{9} a^{4} + \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{4}{9} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{2}{9} a^{6} - \frac{4}{9} a^{5} - \frac{2}{9} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{14} - \frac{1}{27} a^{12} - \frac{1}{27} a^{10} - \frac{1}{27} a^{9} - \frac{1}{9} a^{8} - \frac{4}{27} a^{7} + \frac{1}{3} a^{6} + \frac{4}{27} a^{5} - \frac{8}{27} a^{4} - \frac{2}{27} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{107697800712087} a^{15} - \frac{1192241503975}{107697800712087} a^{14} - \frac{1250532545854}{107697800712087} a^{13} - \frac{5052209855327}{107697800712087} a^{12} - \frac{5666090652565}{107697800712087} a^{11} - \frac{1255614540032}{35899266904029} a^{10} - \frac{223752970505}{107697800712087} a^{9} + \frac{9731843634734}{107697800712087} a^{8} + \frac{13055835776341}{107697800712087} a^{7} - \frac{10144956366083}{107697800712087} a^{6} - \frac{3082074837679}{11966422301343} a^{5} - \frac{15138940078249}{35899266904029} a^{4} + \frac{31008313191371}{107697800712087} a^{3} - \frac{70538308252}{3988807433781} a^{2} + \frac{3980163536065}{11966422301343} a + \frac{221412547275}{1329602477927}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 14545.5858645 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T38):

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 $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-2}) \), 4.2.2704.1, 4.2.10816.1, \(\Q(\sqrt{-2}, \sqrt{13})\), 8.2.1871773696.1, 8.2.1871773696.2, 8.0.1871773696.4

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$