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Properties

Label 16.0.48382396965...7489.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 97^{10}$
Root discriminant $30.22$
Ramified primes $3, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16381, -29141, 33042, -9291, -11672, 11246, 2547, -6786, 1266, 1347, -552, -172, 163, 18, -18, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 18*x^14 + 18*x^13 + 163*x^12 - 172*x^11 - 552*x^10 + 1347*x^9 + 1266*x^8 - 6786*x^7 + 2547*x^6 + 11246*x^5 - 11672*x^4 - 9291*x^3 + 33042*x^2 - 29141*x + 16381)
 
gp: K = bnfinit(x^16 - 2*x^15 - 18*x^14 + 18*x^13 + 163*x^12 - 172*x^11 - 552*x^10 + 1347*x^9 + 1266*x^8 - 6786*x^7 + 2547*x^6 + 11246*x^5 - 11672*x^4 - 9291*x^3 + 33042*x^2 - 29141*x + 16381, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 18 x^{14} + 18 x^{13} + 163 x^{12} - 172 x^{11} - 552 x^{10} + 1347 x^{9} + 1266 x^{8} - 6786 x^{7} + 2547 x^{6} + 11246 x^{5} - 11672 x^{4} - 9291 x^{3} + 33042 x^{2} - 29141 x + 16381 \)

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:  \(483823969655762431707489=3^{8}\cdot 97^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.22$
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:  $3, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{14} a^{14} + \frac{1}{7} a^{13} - \frac{1}{14} a^{12} - \frac{3}{14} a^{11} + \frac{1}{7} a^{10} + \frac{5}{14} a^{9} - \frac{3}{14} a^{8} - \frac{3}{7} a^{7} - \frac{3}{14} a^{6} + \frac{1}{14} a^{5} + \frac{2}{7} a^{4} + \frac{1}{14} a^{3} - \frac{5}{14} a^{2} - \frac{3}{7} a + \frac{3}{14}$, $\frac{1}{65884825562396432490453558187898} a^{15} + \frac{202902982179712455875276115459}{32942412781198216245226779093949} a^{14} + \frac{2589648246856253489303791437564}{32942412781198216245226779093949} a^{13} + \frac{774774339301924719901552129891}{32942412781198216245226779093949} a^{12} - \frac{14907100182843745304194101207802}{32942412781198216245226779093949} a^{11} - \frac{7937112358312501213638935645591}{32942412781198216245226779093949} a^{10} + \frac{3032463934171026718270594095341}{32942412781198216245226779093949} a^{9} - \frac{11001707852410600988211497312836}{32942412781198216245226779093949} a^{8} - \frac{1986849913807768426564692522633}{32942412781198216245226779093949} a^{7} + \frac{5798511916775894723845782809182}{32942412781198216245226779093949} a^{6} + \frac{13931095826349345767620678539584}{32942412781198216245226779093949} a^{5} - \frac{4036796726877004762459406273595}{32942412781198216245226779093949} a^{4} - \frac{15270561786064460884365386343880}{32942412781198216245226779093949} a^{3} + \frac{15566209596819385910674698674290}{32942412781198216245226779093949} a^{2} + \frac{14615551819881079966286567920102}{32942412781198216245226779093949} a - \frac{939693050660498305807418636751}{9412117937485204641493365455414}$

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:  \( \frac{1635044887083493}{25925487728830265582} a^{15} - \frac{2266218916099537}{25925487728830265582} a^{14} - \frac{28982100472406337}{25925487728830265582} a^{13} + \frac{3380228533810795}{12962743864415132791} a^{12} + \frac{243174892866235377}{25925487728830265582} a^{11} - \frac{83579489680520449}{25925487728830265582} a^{10} - \frac{368938399675886743}{12962743864415132791} a^{9} + \frac{1377627882526655127}{25925487728830265582} a^{8} + \frac{2402718408110796251}{25925487728830265582} a^{7} - \frac{3853528833812264616}{12962743864415132791} a^{6} + \frac{248620025616991945}{25925487728830265582} a^{5} + \frac{13011015059914101035}{25925487728830265582} a^{4} - \frac{3673169606597379325}{12962743864415132791} a^{3} - \frac{14729991011466854863}{25925487728830265582} a^{2} + \frac{45273357898236017427}{25925487728830265582} a - \frac{12615760837496252921}{25925487728830265582} \) (order $6$)
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:  \( 329160.9111 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{97}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-291}) \), 4.2.28227.1 x2, 4.0.873.1 x2, \(\Q(\sqrt{-3}, \sqrt{97})\), 8.0.73926513.1, 8.0.695574560817.1, 8.0.7170871761.1

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 8 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.8.6.2$x^{8} + 873 x^{4} + 235225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$