Properties

Label 16.8.135...553.1
Degree $16$
Signature $[8, 4]$
Discriminant $1.358\times 10^{26}$
Root discriminant $42.98$
Ramified primes $17, 83$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 9*x^14 - 146*x^13 + 132*x^12 + 301*x^11 + 933*x^10 + 10461*x^9 - 8365*x^8 - 30971*x^7 - 18709*x^6 + 7320*x^5 + 37149*x^4 + 36402*x^3 + 55167*x^2 + 22519*x - 6323)
 
gp: K = bnfinit(x^16 - 3*x^15 + 9*x^14 - 146*x^13 + 132*x^12 + 301*x^11 + 933*x^10 + 10461*x^9 - 8365*x^8 - 30971*x^7 - 18709*x^6 + 7320*x^5 + 37149*x^4 + 36402*x^3 + 55167*x^2 + 22519*x - 6323, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6323, 22519, 55167, 36402, 37149, 7320, -18709, -30971, -8365, 10461, 933, 301, 132, -146, 9, -3, 1]);
 

\( x^{16} - 3 x^{15} + 9 x^{14} - 146 x^{13} + 132 x^{12} + 301 x^{11} + 933 x^{10} + 10461 x^{9} - 8365 x^{8} - 30971 x^{7} - 18709 x^{6} + 7320 x^{5} + 37149 x^{4} + 36402 x^{3} + 55167 x^{2} + 22519 x - 6323 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(135845792016352372555063553\)\(\medspace = 17^{15}\cdot 83^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $42.98$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $17, 83$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{101027817112911349340933203396635467012540983} a^{15} + \frac{35384239452906246659506338851359451831045699}{101027817112911349340933203396635467012540983} a^{14} - \frac{48828758563064584239141036743459235213066379}{101027817112911349340933203396635467012540983} a^{13} - \frac{43459624113321872946199779191602482477212891}{101027817112911349340933203396635467012540983} a^{12} + \frac{18501582442607918721330400878531056200409319}{101027817112911349340933203396635467012540983} a^{11} + \frac{33362759455347288372156492084765579537763859}{101027817112911349340933203396635467012540983} a^{10} - \frac{4759229989427912665780967855616188913001553}{101027817112911349340933203396635467012540983} a^{9} + \frac{154393955886402745298519661984974523602041}{980852593329236401368283528122674437014961} a^{8} - \frac{6518530649822942737766677755537844172617184}{101027817112911349340933203396635467012540983} a^{7} - \frac{25545804739469019625010217054467144600856650}{101027817112911349340933203396635467012540983} a^{6} - \frac{32675501080098342506612543123426873725732135}{101027817112911349340933203396635467012540983} a^{5} - \frac{25568974840978702654816211751055430574219296}{101027817112911349340933203396635467012540983} a^{4} + \frac{32969735589270881130698693734523369097682484}{101027817112911349340933203396635467012540983} a^{3} - \frac{47163811759127315858414324857090760283610674}{101027817112911349340933203396635467012540983} a^{2} - \frac{25432151693595854740653361788466736821779807}{101027817112911349340933203396635467012540983} a + \frac{16281425127346125838647768658086953988940730}{101027817112911349340933203396635467012540983}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 8827183.62748 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 8827183.62748 \cdot 1}{2\sqrt{135845792016352372555063553}}\approx 0.151087490943$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

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

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
17Data not computed
$83$83.8.0.1$x^{8} - x + 8$$1$$8$$0$$C_8$$[\ ]^{8}$
83.8.4.2$x^{8} - 571787 x^{2} + 1044083062$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$