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Properties

Label 16.0.10708253655...0001.1
Degree $16$
Signature $[0, 8]$
Discriminant $41^{12}\cdot 83^{4}$
Root discriminant $48.91$
Ramified primes $41, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\wr C_4$ (as 16T157)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![368, -3192, 12040, -26348, 38147, -40203, 33239, -23064, 14530, -7841, 3843, -1610, 602, -182, 46, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 46*x^14 - 182*x^13 + 602*x^12 - 1610*x^11 + 3843*x^10 - 7841*x^9 + 14530*x^8 - 23064*x^7 + 33239*x^6 - 40203*x^5 + 38147*x^4 - 26348*x^3 + 12040*x^2 - 3192*x + 368)
 
gp: K = bnfinit(x^16 - 8*x^15 + 46*x^14 - 182*x^13 + 602*x^12 - 1610*x^11 + 3843*x^10 - 7841*x^9 + 14530*x^8 - 23064*x^7 + 33239*x^6 - 40203*x^5 + 38147*x^4 - 26348*x^3 + 12040*x^2 - 3192*x + 368, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 46 x^{14} - 182 x^{13} + 602 x^{12} - 1610 x^{11} + 3843 x^{10} - 7841 x^{9} + 14530 x^{8} - 23064 x^{7} + 33239 x^{6} - 40203 x^{5} + 38147 x^{4} - 26348 x^{3} + 12040 x^{2} - 3192 x + 368 \)

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:  \(1070825365555164876577830001=41^{12}\cdot 83^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.91$
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:  $41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{92} a^{12} - \frac{3}{46} a^{11} + \frac{21}{92} a^{10} + \frac{21}{46} a^{9} + \frac{25}{92} a^{8} + \frac{21}{46} a^{7} - \frac{1}{2} a^{6} - \frac{15}{92} a^{5} - \frac{13}{46} a^{4} - \frac{37}{92} a^{3} + \frac{19}{92} a^{2} - \frac{5}{23} a$, $\frac{1}{92} a^{13} - \frac{15}{92} a^{11} - \frac{4}{23} a^{10} + \frac{1}{92} a^{9} + \frac{2}{23} a^{8} + \frac{11}{46} a^{7} - \frac{15}{92} a^{6} - \frac{6}{23} a^{5} - \frac{9}{92} a^{4} - \frac{19}{92} a^{3} + \frac{1}{46} a^{2} - \frac{7}{23} a$, $\frac{1}{182408216} a^{14} - \frac{7}{182408216} a^{13} + \frac{598473}{182408216} a^{12} - \frac{3590747}{182408216} a^{11} - \frac{27427167}{182408216} a^{10} - \frac{12357367}{182408216} a^{9} - \frac{1034415}{2942068} a^{8} - \frac{73635241}{182408216} a^{7} + \frac{63512005}{182408216} a^{6} - \frac{91142269}{182408216} a^{5} - \frac{4560979}{45602054} a^{4} + \frac{46646871}{182408216} a^{3} + \frac{36451197}{91204108} a^{2} - \frac{18884379}{45602054} a - \frac{322518}{991349}$, $\frac{1}{10762084744} a^{15} + \frac{11}{5381042372} a^{14} - \frac{3666261}{5381042372} a^{13} - \frac{6458008}{1345260593} a^{12} + \frac{554805059}{5381042372} a^{11} - \frac{154253160}{1345260593} a^{10} - \frac{5093733861}{10762084744} a^{9} - \frac{1261378789}{10762084744} a^{8} + \frac{119459911}{2690521186} a^{7} - \frac{1174257005}{2690521186} a^{6} + \frac{5350712901}{10762084744} a^{5} + \frac{3935024451}{10762084744} a^{4} + \frac{531464855}{10762084744} a^{3} - \frac{358568205}{1345260593} a^{2} + \frac{58295623}{1345260593} a - \frac{430881}{58489591}$

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:  \( 11847553.9631 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T157):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.2.394258652003.1 x2, 8.4.32723468116249.2

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$41$41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$