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Properties

Label 16.8.29290697789...7392.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 17^{15}\cdot 19993^{2}$
Root discriminant $69.45$
Ramified primes $2, 17, 19993$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21705584, -8998752, -37194264, -16048352, -3002672, -223744, 310904, -35578, 26519, 17482, -648, 1253, -361, -95, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 9*x^14 - 95*x^13 - 361*x^12 + 1253*x^11 - 648*x^10 + 17482*x^9 + 26519*x^8 - 35578*x^7 + 310904*x^6 - 223744*x^5 - 3002672*x^4 - 16048352*x^3 - 37194264*x^2 - 8998752*x + 21705584)
 
gp: K = bnfinit(x^16 - 3*x^15 + 9*x^14 - 95*x^13 - 361*x^12 + 1253*x^11 - 648*x^10 + 17482*x^9 + 26519*x^8 - 35578*x^7 + 310904*x^6 - 223744*x^5 - 3002672*x^4 - 16048352*x^3 - 37194264*x^2 - 8998752*x + 21705584, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 9 x^{14} - 95 x^{13} - 361 x^{12} + 1253 x^{11} - 648 x^{10} + 17482 x^{9} + 26519 x^{8} - 35578 x^{7} + 310904 x^{6} - 223744 x^{5} - 3002672 x^{4} - 16048352 x^{3} - 37194264 x^{2} - 8998752 x + 21705584 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(292906977896507671746287067392=2^{8}\cdot 17^{15}\cdot 19993^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.45$
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:  $2, 17, 19993$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} + \frac{3}{8} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{886536231835095265656937510162923597074422204020295816} a^{15} - \frac{35402401314981961177443114277493757253036640881454973}{886536231835095265656937510162923597074422204020295816} a^{14} + \frac{60811390566422194873016461449197793737076151723798189}{886536231835095265656937510162923597074422204020295816} a^{13} + \frac{38373992188541418877797266026404415560709709974952063}{886536231835095265656937510162923597074422204020295816} a^{12} - \frac{7915693841281711791244746412766602891945010468211295}{886536231835095265656937510162923597074422204020295816} a^{11} + \frac{18718120687869614251432134837048669449241360079598523}{886536231835095265656937510162923597074422204020295816} a^{10} - \frac{146154811549612896354217650600152067622099369791477293}{443268115917547632828468755081461798537211102010147908} a^{9} + \frac{293729010916378510785550535649802326778060613310431}{1443870084421979260027585521437986314453456358339244} a^{8} - \frac{339241611190951313506742325896369197904978673030756615}{886536231835095265656937510162923597074422204020295816} a^{7} + \frac{12213911589883403475605435594281183070477428297663245}{110817028979386908207117188770365449634302775502536977} a^{6} - \frac{32139839104543712553029412915193945207704344233996487}{443268115917547632828468755081461798537211102010147908} a^{5} - \frac{17061808498770081006493344385924458324054831440665279}{443268115917547632828468755081461798537211102010147908} a^{4} - \frac{52778352847091785292107373017353202153357585156574854}{110817028979386908207117188770365449634302775502536977} a^{3} - \frac{36281193654989282902962107549185731631211653099837988}{110817028979386908207117188770365449634302775502536977} a^{2} + \frac{45578983358274778508321817882152960001790249383723347}{110817028979386908207117188770365449634302775502536977} a + \frac{37938877504621441885302655238438466042263798799954040}{110817028979386908207117188770365449634302775502536977}$

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:  $11$
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:  \( 397510636.446 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n841
Character table for t16n841 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
17Data not computed
19993Data not computed