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Properties

Label 16.0.18000574394...1681.1
Degree $16$
Signature $[0, 8]$
Discriminant $41^{14}\cdot 83^{4}$
Root discriminant $77.80$
Ramified primes $41, 83$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16336, -82952, 340920, -641024, 678907, -418415, 176183, -97086, 54152, -13523, 545, -70, 116, -92, 40, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 40*x^14 - 92*x^13 + 116*x^12 - 70*x^11 + 545*x^10 - 13523*x^9 + 54152*x^8 - 97086*x^7 + 176183*x^6 - 418415*x^5 + 678907*x^4 - 641024*x^3 + 340920*x^2 - 82952*x + 16336)
 
gp: K = bnfinit(x^16 - 6*x^15 + 40*x^14 - 92*x^13 + 116*x^12 - 70*x^11 + 545*x^10 - 13523*x^9 + 54152*x^8 - 97086*x^7 + 176183*x^6 - 418415*x^5 + 678907*x^4 - 641024*x^3 + 340920*x^2 - 82952*x + 16336, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 40 x^{14} - 92 x^{13} + 116 x^{12} - 70 x^{11} + 545 x^{10} - 13523 x^{9} + 54152 x^{8} - 97086 x^{7} + 176183 x^{6} - 418415 x^{5} + 678907 x^{4} - 641024 x^{3} + 340920 x^{2} - 82952 x + 16336 \)

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:  \(1800057439498232157527332231681=41^{14}\cdot 83^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.80$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{664} a^{14} - \frac{23}{664} a^{13} + \frac{43}{664} a^{12} + \frac{133}{664} a^{11} + \frac{95}{664} a^{10} - \frac{169}{664} a^{9} + \frac{45}{332} a^{8} + \frac{55}{664} a^{7} + \frac{37}{664} a^{6} - \frac{199}{664} a^{5} - \frac{63}{332} a^{4} - \frac{89}{664} a^{3} + \frac{59}{166} a^{2} + \frac{11}{166} a - \frac{8}{83}$, $\frac{1}{1900062359034945964772707067279971144} a^{15} - \frac{864879324118852242912606748188045}{1900062359034945964772707067279971144} a^{14} - \frac{25773263081935673553564920559277307}{1900062359034945964772707067279971144} a^{13} + \frac{147622217483233434018962460496125885}{1900062359034945964772707067279971144} a^{12} - \frac{328375812852838752037589119544516131}{1900062359034945964772707067279971144} a^{11} - \frac{16036588062236344459300362348422893}{1900062359034945964772707067279971144} a^{10} + \frac{95822682806198939375149278117426990}{237507794879368245596588383409996393} a^{9} + \frac{898917571033000021342942105150229997}{1900062359034945964772707067279971144} a^{8} + \frac{464748647197687065906738078583845347}{1900062359034945964772707067279971144} a^{7} - \frac{879703399682613199762432144586168541}{1900062359034945964772707067279971144} a^{6} + \frac{327899682391594593814340020274664313}{950031179517472982386353533639985572} a^{5} - \frac{705623915279630345442149587680893101}{1900062359034945964772707067279971144} a^{4} + \frac{22479805536738078478839610215288282}{237507794879368245596588383409996393} a^{3} + \frac{350005531686444893645088770337255175}{950031179517472982386353533639985572} a^{2} - \frac{169633774104390979895043448628524177}{475015589758736491193176766819992786} a - \frac{63592755065835673738205142949453}{232622717805453717528490091488733}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 93875598.4999 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 4.2.139523.2, 4.2.5720443.1, 8.4.1341662192766209.3, 8.0.194754273881.1, 8.4.32723468116249.1

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 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
$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}$