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Properties

Label 16.16.1364873065...2944.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{48}\cdot 449^{2}\cdot 1889^{5}$
Root discriminant $181.32$
Ramified primes $2, 449, 1889$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1186

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-573429908638, -1378932819352, -762305657052, 217601183232, 180053223426, -11380691704, -13786393584, 289793200, 498003651, -4234648, -9438176, 32376, 95858, -96, -492, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 492*x^14 - 96*x^13 + 95858*x^12 + 32376*x^11 - 9438176*x^10 - 4234648*x^9 + 498003651*x^8 + 289793200*x^7 - 13786393584*x^6 - 11380691704*x^5 + 180053223426*x^4 + 217601183232*x^3 - 762305657052*x^2 - 1378932819352*x - 573429908638)
 
gp: K = bnfinit(x^16 - 492*x^14 - 96*x^13 + 95858*x^12 + 32376*x^11 - 9438176*x^10 - 4234648*x^9 + 498003651*x^8 + 289793200*x^7 - 13786393584*x^6 - 11380691704*x^5 + 180053223426*x^4 + 217601183232*x^3 - 762305657052*x^2 - 1378932819352*x - 573429908638, 1)
 

Normalized defining polynomial

\( x^{16} - 492 x^{14} - 96 x^{13} + 95858 x^{12} + 32376 x^{11} - 9438176 x^{10} - 4234648 x^{9} + 498003651 x^{8} + 289793200 x^{7} - 13786393584 x^{6} - 11380691704 x^{5} + 180053223426 x^{4} + 217601183232 x^{3} - 762305657052 x^{2} - 1378932819352 x - 573429908638 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1364873065607290683946275468103122944=2^{48}\cdot 449^{2}\cdot 1889^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $181.32$
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:  $2, 449, 1889$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{449} a^{12} - \frac{217}{449} a^{11} + \frac{159}{449} a^{10} + \frac{28}{449} a^{9} + \frac{196}{449} a^{8} + \frac{54}{449} a^{7} + \frac{45}{449} a^{6} - \frac{79}{449} a^{5} - \frac{46}{449} a^{4} + \frac{124}{449} a^{3} + \frac{147}{449} a^{2} + \frac{68}{449} a - \frac{51}{449}$, $\frac{1}{449} a^{13} + \frac{215}{449} a^{11} - \frac{42}{449} a^{10} - \frac{14}{449} a^{9} - \frac{69}{449} a^{8} + \frac{89}{449} a^{7} - \frac{192}{449} a^{6} - \frac{127}{449} a^{5} + \frac{20}{449} a^{4} + \frac{115}{449} a^{3} + \frac{88}{449} a^{2} - \frac{112}{449} a + \frac{158}{449}$, $\frac{1}{449} a^{14} - \frac{83}{449} a^{11} - \frac{75}{449} a^{10} + \frac{197}{449} a^{9} + \frac{155}{449} a^{8} - \frac{128}{449} a^{7} + \frac{76}{449} a^{6} - \frac{57}{449} a^{5} + \frac{127}{449} a^{4} - \frac{81}{449} a^{3} + \frac{162}{449} a^{2} - \frac{94}{449} a + \frac{189}{449}$, $\frac{1}{488699074273803679170942445434994638438943993602064046838453548395588745041} a^{15} - \frac{353085896296137833398840262454363501203927689086961087132603582834694101}{488699074273803679170942445434994638438943993602064046838453548395588745041} a^{14} + \frac{97559163956307224121686609882201150086489884393041605714274707261934092}{488699074273803679170942445434994638438943993602064046838453548395588745041} a^{13} + \frac{259940247095323003877048087875645620333362949698130627869216961499757976}{488699074273803679170942445434994638438943993602064046838453548395588745041} a^{12} - \frac{71053370655409370102387187390144118275053689309458162372758935404417458016}{488699074273803679170942445434994638438943993602064046838453548395588745041} a^{11} - \frac{205964000211454416822870259828327282882584510313305962927692189309822873358}{488699074273803679170942445434994638438943993602064046838453548395588745041} a^{10} - \frac{238716491430232625077851131126082793961158585034774138626289475504278793110}{488699074273803679170942445434994638438943993602064046838453548395588745041} a^{9} + \frac{45702515615161767691899007292705786686788762273619566475279499699010440379}{488699074273803679170942445434994638438943993602064046838453548395588745041} a^{8} + \frac{178097986048553117447988889333583849051284988765290457140044621814323669874}{488699074273803679170942445434994638438943993602064046838453548395588745041} a^{7} - \frac{164531080817696230051554256551770498744776600946058912568375113140430509595}{488699074273803679170942445434994638438943993602064046838453548395588745041} a^{6} + \frac{83736548661550702376634863750553589066109763091667743546529936243854826}{2191475669389254166685840562488765194793470823327641465643289454688738767} a^{5} + \frac{47643892649153353580372151802653396567964846702887296851587679369895425405}{488699074273803679170942445434994638438943993602064046838453548395588745041} a^{4} + \frac{37084247006105209574760125580677208340360099409211587097054808924003909712}{488699074273803679170942445434994638438943993602064046838453548395588745041} a^{3} + \frac{148192236125597095604208148771106064960045781357234131489296096204544705575}{488699074273803679170942445434994638438943993602064046838453548395588745041} a^{2} + \frac{197686820532420205872601741572491182491363810247566226480072868876340091081}{488699074273803679170942445434994638438943993602064046838453548395588745041} a + \frac{94605664312862484820671450462647333788213951947369334206266534113042329122}{488699074273803679170942445434994638438943993602064046838453548395588745041}$

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:  $15$
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:  \( 24665861011400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1186:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 46 conjugacy class representatives for t16n1186
Character table for t16n1186 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7923040256.1

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 R $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
449Data not computed
1889Data not computed