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Properties

Label 16.16.8469394764...6001.1
Degree $16$
Signature $[16, 0]$
Discriminant $13^{14}\cdot 173^{14}$
Root discriminant $857.02$
Ramified primes $13, 173$
Class number $50$ (GRH)
Class group $[5, 10]$ (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![-151797202614301, -451442241263186, -102338147908928, 39953039198538, 10246556574051, -1243559750672, -369548628666, 17009155785, 6520100240, -104127446, -62605754, 207407, 333504, 330, -919, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 919*x^14 + 330*x^13 + 333504*x^12 + 207407*x^11 - 62605754*x^10 - 104127446*x^9 + 6520100240*x^8 + 17009155785*x^7 - 369548628666*x^6 - 1243559750672*x^5 + 10246556574051*x^4 + 39953039198538*x^3 - 102338147908928*x^2 - 451442241263186*x - 151797202614301)
 
gp: K = bnfinit(x^16 - x^15 - 919*x^14 + 330*x^13 + 333504*x^12 + 207407*x^11 - 62605754*x^10 - 104127446*x^9 + 6520100240*x^8 + 17009155785*x^7 - 369548628666*x^6 - 1243559750672*x^5 + 10246556574051*x^4 + 39953039198538*x^3 - 102338147908928*x^2 - 451442241263186*x - 151797202614301, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 919 x^{14} + 330 x^{13} + 333504 x^{12} + 207407 x^{11} - 62605754 x^{10} - 104127446 x^{9} + 6520100240 x^{8} + 17009155785 x^{7} - 369548628666 x^{6} - 1243559750672 x^{5} + 10246556574051 x^{4} + 39953039198538 x^{3} - 102338147908928 x^{2} - 451442241263186 x - 151797202614301 \)

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:  \(84693947642368239934810426203628826170679406001=13^{14}\cdot 173^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $857.02$
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:  $13, 173$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} + \frac{4}{13} a^{10} + \frac{1}{13} a^{9} + \frac{6}{13} a^{8} + \frac{5}{13} a^{7} + \frac{6}{13} a^{5} + \frac{5}{13} a^{4} + \frac{2}{13} a^{3} - \frac{3}{13} a^{2} + \frac{2}{13} a - \frac{3}{13}$, $\frac{1}{13} a^{12} - \frac{2}{13} a^{10} + \frac{2}{13} a^{9} - \frac{6}{13} a^{8} + \frac{6}{13} a^{7} + \frac{6}{13} a^{6} - \frac{6}{13} a^{5} - \frac{5}{13} a^{4} + \frac{2}{13} a^{3} + \frac{1}{13} a^{2} + \frac{2}{13} a - \frac{1}{13}$, $\frac{1}{13} a^{13} - \frac{3}{13} a^{10} - \frac{4}{13} a^{9} + \frac{5}{13} a^{8} + \frac{3}{13} a^{7} - \frac{6}{13} a^{6} - \frac{6}{13} a^{5} - \frac{1}{13} a^{4} + \frac{5}{13} a^{3} - \frac{4}{13} a^{2} + \frac{3}{13} a - \frac{6}{13}$, $\frac{1}{43433} a^{14} - \frac{1598}{43433} a^{13} - \frac{853}{43433} a^{12} - \frac{1650}{43433} a^{11} - \frac{11961}{43433} a^{10} - \frac{19953}{43433} a^{9} - \frac{2806}{43433} a^{8} - \frac{6765}{43433} a^{7} - \frac{14308}{43433} a^{6} - \frac{1523}{3341} a^{5} + \frac{2508}{43433} a^{4} + \frac{1254}{43433} a^{3} - \frac{6248}{43433} a^{2} + \frac{20828}{43433} a + \frac{11664}{43433}$, $\frac{1}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573} a^{15} + \frac{317828308092814374273648229414982687708204014548581246718681088289857150011736}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573} a^{14} + \frac{1153884923589773315445803196521479493609843973235587017419160891409959830976010289}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573} a^{13} - \frac{364807994368763388794039862078591282481792715844673739245173018062750239034618959}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573} a^{12} - \frac{446467508300257026579067961278322475611449414811261496646509800214782525603494994}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573} a^{11} - \frac{11321258162325868736122368722659885175401923301229093533097215290986284228025873473}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573} a^{10} + \frac{38308412428421823132864948558795727378860048315376685809512541155100820220993599}{719463847582658798505191241518999072227991773738990483593585367045445905684009711} a^{9} - \frac{254922775091543165866689704791239186893397809959475065836461846298118131966253039}{2379765034311871410440247952716689238907972790059737753424936214073397995724032121} a^{8} - \frac{7171038166566618040097735081982831311247533098951148862029649519935078307383060625}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573} a^{7} + \frac{12199211818829251038704026504434440080466926255712701983491759997144744818805248970}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573} a^{6} + \frac{6361137641451567287969098767031462464558275622951539090865629320231243034218328870}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573} a^{5} + \frac{3180209883806310630433791370408886251714411469045933897549783341502921091869007664}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573} a^{4} - \frac{15347098675565893350864218433187197033306031900762907756308220488238663403489407888}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573} a^{3} - \frac{10857542594920385898054203339301877073771374401500870390935552236115908998903376805}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573} a^{2} + \frac{2306534357640814564134467259859297262889046890379654757627460890564402251047757673}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573} a - \frac{2837318861523392343669916056302733185508210838204042906621190210733654836522177110}{30936945446054328335723223385316960105803646270776590794524170782954173944412417573}$

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

Class group and class number

$C_{5}\times C_{10}$, which has order $50$ (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:  \( 30374024111100000 \) (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

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{2249}) \), \(\Q(\sqrt{173}) \), \(\Q(\sqrt{13}, \sqrt{173})\), 8.8.129400731862107174001.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$173$173.8.7.2$x^{8} - 692$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
173.8.7.2$x^{8} - 692$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$