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Properties

Label 16.0.29464066849...2241.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{14}\cdot 83^{6}$
Root discriminant $123.55$
Ramified primes $37, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3528539, 7064925, -303276, -102100, 4953545, -2938105, 295915, -41802, 186709, -63940, -1085, 3641, -475, 78, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 5*x^14 + 78*x^13 - 475*x^12 + 3641*x^11 - 1085*x^10 - 63940*x^9 + 186709*x^8 - 41802*x^7 + 295915*x^6 - 2938105*x^5 + 4953545*x^4 - 102100*x^3 - 303276*x^2 + 7064925*x + 3528539)
 
gp: K = bnfinit(x^16 - 2*x^15 - 5*x^14 + 78*x^13 - 475*x^12 + 3641*x^11 - 1085*x^10 - 63940*x^9 + 186709*x^8 - 41802*x^7 + 295915*x^6 - 2938105*x^5 + 4953545*x^4 - 102100*x^3 - 303276*x^2 + 7064925*x + 3528539, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 5 x^{14} + 78 x^{13} - 475 x^{12} + 3641 x^{11} - 1085 x^{10} - 63940 x^{9} + 186709 x^{8} - 41802 x^{7} + 295915 x^{6} - 2938105 x^{5} + 4953545 x^{4} - 102100 x^{3} - 303276 x^{2} + 7064925 x + 3528539 \)

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:  \(2946406684936922037860575506582241=37^{14}\cdot 83^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.55$
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:  $37, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{49} a^{11} - \frac{1}{49} a^{10} + \frac{1}{49} a^{9} + \frac{2}{49} a^{8} + \frac{2}{49} a^{7} + \frac{1}{49} a^{6} + \frac{17}{49} a^{5} - \frac{18}{49} a^{4} + \frac{16}{49} a^{2} - \frac{3}{7} a$, $\frac{1}{49} a^{12} + \frac{3}{49} a^{9} - \frac{3}{49} a^{8} + \frac{3}{49} a^{7} - \frac{3}{49} a^{6} + \frac{13}{49} a^{5} - \frac{11}{49} a^{4} - \frac{5}{49} a^{3} + \frac{16}{49} a^{2} - \frac{2}{7} a$, $\frac{1}{49} a^{13} + \frac{3}{49} a^{10} - \frac{3}{49} a^{9} + \frac{3}{49} a^{8} - \frac{3}{49} a^{7} - \frac{1}{49} a^{6} - \frac{18}{49} a^{5} + \frac{16}{49} a^{4} + \frac{2}{49} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{343} a^{14} + \frac{2}{343} a^{13} - \frac{1}{343} a^{12} + \frac{3}{343} a^{11} + \frac{3}{343} a^{10} - \frac{13}{343} a^{9} + \frac{13}{343} a^{8} - \frac{10}{343} a^{7} + \frac{11}{343} a^{6} - \frac{68}{343} a^{5} + \frac{101}{343} a^{4} - \frac{75}{343} a^{3} + \frac{166}{343} a^{2} - \frac{5}{49} a - \frac{3}{7}$, $\frac{1}{20797168317419612357686405738782028472409089} a^{15} + \frac{3009868475597793168956687297556061882558}{2971024045345658908240915105540289781772727} a^{14} + \frac{181665854476536240713221957956523293800651}{20797168317419612357686405738782028472409089} a^{13} + \frac{105830469723359017137572129035526217017200}{20797168317419612357686405738782028472409089} a^{12} - \frac{206866992471510742881082271062513401177425}{20797168317419612357686405738782028472409089} a^{11} + \frac{1157085247042036711603217086880306874954225}{20797168317419612357686405738782028472409089} a^{10} - \frac{888137081967886273108721345347381682968589}{20797168317419612357686405738782028472409089} a^{9} - \frac{1333121936537465735733843374397062125158975}{20797168317419612357686405738782028472409089} a^{8} + \frac{211463414234814924460897745287956180913649}{20797168317419612357686405738782028472409089} a^{7} + \frac{1327968667011365430223678922089483191787329}{20797168317419612357686405738782028472409089} a^{6} - \frac{2169895897491515032108313766720707822334474}{20797168317419612357686405738782028472409089} a^{5} - \frac{5619883100770549451069203886730258772215503}{20797168317419612357686405738782028472409089} a^{4} + \frac{7577453741965765230616389017247681238557369}{20797168317419612357686405738782028472409089} a^{3} - \frac{2862601852787358971088235167764624268615508}{20797168317419612357686405738782028472409089} a^{2} + \frac{192988334401282036660443591343332531653774}{2971024045345658908240915105540289781772727} a - \frac{358454689450270441585913067130715373778}{424432006477951272605845015077184254538961}$

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:  \( 87394636266.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

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 $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{37}) \), 4.0.50653.1, 4.2.113627.1, 4.2.4204199.1, 8.0.653985701569237.1 x2, 8.0.17675289231601.1

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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
37Data not computed
83Data not computed