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Properties

Label 16.0.15813464473...6121.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 29^{14}$
Root discriminant $43.39$
Ramified primes $3, 29$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![77917, -251193, 404877, -465259, 417254, -291930, 186225, -106947, 52719, -24016, 9477, -2928, 786, -206, 39, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 39*x^14 - 206*x^13 + 786*x^12 - 2928*x^11 + 9477*x^10 - 24016*x^9 + 52719*x^8 - 106947*x^7 + 186225*x^6 - 291930*x^5 + 417254*x^4 - 465259*x^3 + 404877*x^2 - 251193*x + 77917)
 
gp: K = bnfinit(x^16 - 5*x^15 + 39*x^14 - 206*x^13 + 786*x^12 - 2928*x^11 + 9477*x^10 - 24016*x^9 + 52719*x^8 - 106947*x^7 + 186225*x^6 - 291930*x^5 + 417254*x^4 - 465259*x^3 + 404877*x^2 - 251193*x + 77917, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 39 x^{14} - 206 x^{13} + 786 x^{12} - 2928 x^{11} + 9477 x^{10} - 24016 x^{9} + 52719 x^{8} - 106947 x^{7} + 186225 x^{6} - 291930 x^{5} + 417254 x^{4} - 465259 x^{3} + 404877 x^{2} - 251193 x + 77917 \)

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:  \(158134644731459542671806121=3^{12}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.39$
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:  $3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{35} a^{8} + \frac{1}{35} a^{7} + \frac{1}{35} a^{6} - \frac{17}{35} a^{5} + \frac{2}{35} a^{4} - \frac{13}{35} a^{3} - \frac{11}{35} a^{2} - \frac{6}{35} a - \frac{1}{5}$, $\frac{1}{35} a^{9} + \frac{2}{35} a^{6} - \frac{6}{35} a^{5} - \frac{2}{7} a^{4} - \frac{13}{35} a^{3} + \frac{3}{7} a^{2} + \frac{4}{35} a + \frac{1}{5}$, $\frac{1}{35} a^{10} + \frac{2}{35} a^{7} - \frac{1}{35} a^{6} + \frac{2}{7} a^{5} - \frac{3}{35} a^{4} - \frac{3}{7} a^{3} - \frac{11}{35} a^{2} + \frac{17}{35} a$, $\frac{1}{245} a^{11} + \frac{1}{245} a^{10} - \frac{1}{245} a^{9} - \frac{3}{245} a^{8} + \frac{6}{245} a^{7} + \frac{12}{245} a^{6} - \frac{72}{245} a^{5} - \frac{103}{245} a^{4} + \frac{27}{245} a^{3} - \frac{19}{245} a^{2} + \frac{88}{245} a + \frac{9}{35}$, $\frac{1}{490} a^{12} - \frac{1}{490} a^{11} - \frac{3}{490} a^{10} - \frac{1}{490} a^{9} - \frac{1}{245} a^{8} - \frac{1}{35} a^{7} - \frac{1}{98} a^{6} + \frac{209}{490} a^{5} - \frac{15}{98} a^{4} + \frac{107}{245} a^{3} - \frac{1}{14} a^{2} - \frac{32}{245} a + \frac{31}{70}$, $\frac{1}{3430} a^{13} + \frac{3}{3430} a^{12} - \frac{1}{490} a^{11} - \frac{13}{3430} a^{10} - \frac{24}{1715} a^{9} + \frac{2}{343} a^{8} + \frac{121}{3430} a^{7} + \frac{31}{490} a^{6} + \frac{1069}{3430} a^{5} - \frac{211}{1715} a^{4} + \frac{1381}{3430} a^{3} - \frac{18}{1715} a^{2} + \frac{521}{3430} a - \frac{24}{49}$, $\frac{1}{120050} a^{14} - \frac{1}{17150} a^{13} - \frac{3}{12005} a^{12} + \frac{19}{12005} a^{11} - \frac{1269}{120050} a^{10} - \frac{29}{4802} a^{9} - \frac{611}{120050} a^{8} - \frac{5739}{120050} a^{7} + \frac{368}{12005} a^{6} + \frac{1993}{24010} a^{5} - \frac{3796}{12005} a^{4} + \frac{15}{343} a^{3} - \frac{11498}{60025} a^{2} + \frac{22434}{60025} a + \frac{1917}{17150}$, $\frac{1}{114885699384802229450} a^{15} + \frac{307670574342689}{114885699384802229450} a^{14} + \frac{12368646625888803}{114885699384802229450} a^{13} - \frac{10144358603436371}{22977139876960445890} a^{12} + \frac{49493463562973498}{57442849692401114725} a^{11} + \frac{5417553770172814}{8206121384628730675} a^{10} - \frac{978259287453761811}{114885699384802229450} a^{9} + \frac{249109377055958903}{22977139876960445890} a^{8} + \frac{1557619566623751711}{114885699384802229450} a^{7} - \frac{256112856841341873}{11488569938480222945} a^{6} + \frac{111489083595259269}{656489710770298454} a^{5} - \frac{522469086272160862}{2297713987696044589} a^{4} - \frac{24667187771492790791}{114885699384802229450} a^{3} - \frac{1669921532411300552}{8206121384628730675} a^{2} + \frac{6368315615575620331}{57442849692401114725} a - \frac{346857635774198814}{8206121384628730675}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{134184973308}{4567293447753925} a^{15} - \frac{614211344462}{4567293447753925} a^{14} + \frac{4944214085347}{4567293447753925} a^{13} - \frac{5127613834923}{913458689550785} a^{12} + \frac{93823257782803}{4567293447753925} a^{11} - \frac{50545014700868}{652470492536275} a^{10} + \frac{1120307305434862}{4567293447753925} a^{9} - \frac{2733229551636586}{4567293447753925} a^{8} + \frac{5925367268735239}{4567293447753925} a^{7} - \frac{2372076330121971}{913458689550785} a^{6} + \frac{562356495654404}{130494098507255} a^{5} - \frac{6136203574088692}{913458689550785} a^{4} + \frac{43221464880367017}{4567293447753925} a^{3} - \frac{1222436880682859}{130494098507255} a^{2} + \frac{37099042432593054}{4567293447753925} a - \frac{2100625914697357}{652470492536275} \) (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:  \( 20355392.4722 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-87}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{29})\), 4.4.219501.1, 4.0.24389.1, 8.0.48180689001.1, 8.4.12575159829261.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
29Data not computed