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Properties

Label 16.4.25605559374...1477.4
Degree $16$
Signature $[4, 6]$
Discriminant $13^{15}\cdot 29^{8}$
Root discriminant $59.64$
Ramified primes $13, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1688688, -4861080, 5367456, -2798238, 298753, 561171, -254072, 4836, 1677, 5148, -715, 156, -39, -66, 16, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 16*x^14 - 66*x^13 - 39*x^12 + 156*x^11 - 715*x^10 + 5148*x^9 + 1677*x^8 + 4836*x^7 - 254072*x^6 + 561171*x^5 + 298753*x^4 - 2798238*x^3 + 5367456*x^2 - 4861080*x + 1688688)
 
gp: K = bnfinit(x^16 - 3*x^15 + 16*x^14 - 66*x^13 - 39*x^12 + 156*x^11 - 715*x^10 + 5148*x^9 + 1677*x^8 + 4836*x^7 - 254072*x^6 + 561171*x^5 + 298753*x^4 - 2798238*x^3 + 5367456*x^2 - 4861080*x + 1688688, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 16 x^{14} - 66 x^{13} - 39 x^{12} + 156 x^{11} - 715 x^{10} + 5148 x^{9} + 1677 x^{8} + 4836 x^{7} - 254072 x^{6} + 561171 x^{5} + 298753 x^{4} - 2798238 x^{3} + 5367456 x^{2} - 4861080 x + 1688688 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(25605559374504409818155101477=13^{15}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.64$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{78} a^{8} + \frac{5}{78} a^{7} + \frac{1}{39} a^{6} - \frac{17}{78} a^{5} + \frac{31}{78} a^{4} - \frac{2}{39} a^{3} + \frac{1}{39} a^{2} + \frac{31}{78} a - \frac{2}{13}$, $\frac{1}{78} a^{9} + \frac{1}{26} a^{7} + \frac{2}{13} a^{6} + \frac{2}{13} a^{5} - \frac{1}{26} a^{4} - \frac{5}{13} a^{3} + \frac{7}{26} a^{2} + \frac{1}{39} a - \frac{3}{13}$, $\frac{1}{156} a^{10} - \frac{1}{156} a^{8} - \frac{2}{39} a^{7} - \frac{35}{156} a^{6} + \frac{1}{6} a^{5} - \frac{37}{156} a^{4} + \frac{19}{39} a^{3} - \frac{15}{52} a^{2} + \frac{7}{78} a + \frac{4}{13}$, $\frac{1}{156} a^{11} - \frac{1}{156} a^{9} + \frac{5}{156} a^{7} - \frac{3}{13} a^{6} - \frac{17}{156} a^{5} + \frac{1}{13} a^{4} - \frac{77}{156} a^{3} + \frac{5}{26} a^{2} + \frac{31}{78} a + \frac{5}{13}$, $\frac{1}{156} a^{12} - \frac{1}{13} a^{7} + \frac{3}{26} a^{6} - \frac{2}{13} a^{5} - \frac{1}{39} a^{4} - \frac{5}{13} a^{3} - \frac{23}{52} a^{2} + \frac{9}{26} a - \frac{5}{13}$, $\frac{1}{468} a^{13} + \frac{1}{468} a^{12} - \frac{1}{468} a^{11} - \frac{1}{468} a^{10} - \frac{1}{468} a^{9} - \frac{1}{468} a^{8} + \frac{1}{468} a^{7} - \frac{17}{468} a^{6} - \frac{101}{468} a^{5} + \frac{43}{468} a^{4} + \frac{37}{234} a^{3} + \frac{14}{117} a^{2} - \frac{1}{13} a + \frac{6}{13}$, $\frac{1}{1404} a^{14} + \frac{1}{1404} a^{13} - \frac{1}{1404} a^{12} - \frac{1}{1404} a^{11} - \frac{1}{351} a^{10} - \frac{1}{1404} a^{9} + \frac{1}{351} a^{8} - \frac{71}{1404} a^{7} + \frac{119}{702} a^{6} + \frac{277}{1404} a^{5} - \frac{283}{1404} a^{4} + \frac{113}{351} a^{3} - \frac{67}{156} a^{2} + \frac{7}{39} a + \frac{3}{13}$, $\frac{1}{24318731556172021000105318813776} a^{15} + \frac{854753425656017633791007557}{8106243852057340333368439604592} a^{14} + \frac{11106940209030195935780571101}{12159365778086010500052659406888} a^{13} - \frac{9260433015994634664960453139}{4053121926028670166684219802296} a^{12} - \frac{21466997577246223128580240633}{8106243852057340333368439604592} a^{11} + \frac{10925492339557057761292958051}{4053121926028670166684219802296} a^{10} + \frac{66518322792202546424028280337}{24318731556172021000105318813776} a^{9} - \frac{556436327828283067822895581}{4053121926028670166684219802296} a^{8} - \frac{412636219491747417463100547533}{8106243852057340333368439604592} a^{7} + \frac{286611147094748860984980596075}{4053121926028670166684219802296} a^{6} + \frac{890123716333615705085182669465}{6079682889043005250026329703444} a^{5} + \frac{164438162908622634568887211253}{900693761339704481485382178288} a^{4} + \frac{764466654137505952406283219343}{1870671658167078538469639908752} a^{3} - \frac{489589586757462092664229035619}{1013280481507167541671054950574} a^{2} - \frac{113810517776383139107505031211}{337760160502389180557018316858} a - \frac{17126513242021404921653442181}{112586720167463060185672772286}$

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:  $9$
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:  \( 190310943.764 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.63713.1, 8.4.52771502797.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ $16$

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
13Data not computed
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$