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Properties

Label 16.0.18161513224...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 29^{14}$
Root discriminant $77.84$
Ramified primes $5, 29$
Class number $656$ (GRH)
Class group $[2, 2, 164]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![957431, 1340385, 2107098, 836180, 798933, 83880, 114546, -4545, 19125, -85, 2936, -520, 218, -20, 23, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 23*x^14 - 20*x^13 + 218*x^12 - 520*x^11 + 2936*x^10 - 85*x^9 + 19125*x^8 - 4545*x^7 + 114546*x^6 + 83880*x^5 + 798933*x^4 + 836180*x^3 + 2107098*x^2 + 1340385*x + 957431)
 
gp: K = bnfinit(x^16 - 5*x^15 + 23*x^14 - 20*x^13 + 218*x^12 - 520*x^11 + 2936*x^10 - 85*x^9 + 19125*x^8 - 4545*x^7 + 114546*x^6 + 83880*x^5 + 798933*x^4 + 836180*x^3 + 2107098*x^2 + 1340385*x + 957431, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 23 x^{14} - 20 x^{13} + 218 x^{12} - 520 x^{11} + 2936 x^{10} - 85 x^{9} + 19125 x^{8} - 4545 x^{7} + 114546 x^{6} + 83880 x^{5} + 798933 x^{4} + 836180 x^{3} + 2107098 x^{2} + 1340385 x + 957431 \)

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:  \(1816151322484127584722900390625=5^{14}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.84$
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:  $5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{179} a^{14} - \frac{15}{179} a^{13} + \frac{15}{179} a^{12} + \frac{52}{179} a^{11} + \frac{13}{179} a^{10} + \frac{84}{179} a^{9} + \frac{42}{179} a^{8} + \frac{6}{179} a^{7} + \frac{78}{179} a^{6} - \frac{8}{179} a^{5} - \frac{86}{179} a^{4} + \frac{84}{179} a^{3} - \frac{84}{179} a^{2} - \frac{10}{179} a + \frac{36}{179}$, $\frac{1}{4828872109685134798401513288141061931521849} a^{15} - \frac{5509620048908734349701714001830494688128}{4828872109685134798401513288141061931521849} a^{14} + \frac{1254855079381517598041709327218756264730806}{4828872109685134798401513288141061931521849} a^{13} - \frac{2300870990034022960882598907668508387449268}{4828872109685134798401513288141061931521849} a^{12} - \frac{2030034007858068200996106890019768165556225}{4828872109685134798401513288141061931521849} a^{11} + \frac{1473165800678781586055153203808489468930537}{4828872109685134798401513288141061931521849} a^{10} - \frac{33269460780306078513200875694097831602693}{68012283235001898569035398424521999035519} a^{9} - \frac{1311559789963641780077035995558154936785651}{4828872109685134798401513288141061931521849} a^{8} - \frac{1333571651192086424259913992921774301428807}{4828872109685134798401513288141061931521849} a^{7} + \frac{701458556529694993960741668439299088647271}{4828872109685134798401513288141061931521849} a^{6} - \frac{56996965839457758707473404724850647798760}{4828872109685134798401513288141061931521849} a^{5} + \frac{773106449116925892210922997065461174078947}{4828872109685134798401513288141061931521849} a^{4} - \frac{2332021948347679760549701862659588890583373}{4828872109685134798401513288141061931521849} a^{3} + \frac{1502068795538019109009818326914059969603261}{4828872109685134798401513288141061931521849} a^{2} - \frac{2017594782219503789710525674842407315460960}{4828872109685134798401513288141061931521849} a + \frac{1500288047959500770660070181983376388414895}{4828872109685134798401513288141061931521849}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{164}$, which has order $656$ (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:  \( 499585.18112 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

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 $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.9294114390625.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$29$29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$