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Properties

Label 16.4.25893015832...017.16
Degree $16$
Signature $[4, 6]$
Discriminant $17^{15}\cdot 67^{6}$
Root discriminant $68.92$
Ramified primes $17, 67$
Class number $64$ (GRH)
Class group $[2, 2, 4, 4]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9493447, -64012350, -11710839, -18441153, -20644371, -3692832, -2486412, -478326, 10386, -44177, 25243, -3558, 2189, -163, 77, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 77*x^14 - 163*x^13 + 2189*x^12 - 3558*x^11 + 25243*x^10 - 44177*x^9 + 10386*x^8 - 478326*x^7 - 2486412*x^6 - 3692832*x^5 - 20644371*x^4 - 18441153*x^3 - 11710839*x^2 - 64012350*x + 9493447)
 
gp: K = bnfinit(x^16 - 3*x^15 + 77*x^14 - 163*x^13 + 2189*x^12 - 3558*x^11 + 25243*x^10 - 44177*x^9 + 10386*x^8 - 478326*x^7 - 2486412*x^6 - 3692832*x^5 - 20644371*x^4 - 18441153*x^3 - 11710839*x^2 - 64012350*x + 9493447, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 77 x^{14} - 163 x^{13} + 2189 x^{12} - 3558 x^{11} + 25243 x^{10} - 44177 x^{9} + 10386 x^{8} - 478326 x^{7} - 2486412 x^{6} - 3692832 x^{5} - 20644371 x^{4} - 18441153 x^{3} - 11710839 x^{2} - 64012350 x + 9493447 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(258930158322830089457985795017=17^{15}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.92$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{6294659757359515833476913032043701676510023341657731843901} a^{15} - \frac{748756703477750235893695970124403245660643600644766415412}{6294659757359515833476913032043701676510023341657731843901} a^{14} + \frac{1531861382967947923462319233037863562729593548013363792449}{6294659757359515833476913032043701676510023341657731843901} a^{13} - \frac{1811181852776962576115647341436375396333255931783721481301}{6294659757359515833476913032043701676510023341657731843901} a^{12} + \frac{506985196668408052188537728011622742086665445760953707899}{6294659757359515833476913032043701676510023341657731843901} a^{11} - \frac{2474736776528509582252799020997600494480330918670234002829}{6294659757359515833476913032043701676510023341657731843901} a^{10} - \frac{2608252805631527498781977174977110097910946918570049155290}{6294659757359515833476913032043701676510023341657731843901} a^{9} + \frac{1350104875583651182908827927873413658900383639883681757262}{6294659757359515833476913032043701676510023341657731843901} a^{8} - \frac{2085290781281019947832671007414488383944373298180544000467}{6294659757359515833476913032043701676510023341657731843901} a^{7} + \frac{359461574410492900717104179758020232133573728671413288363}{6294659757359515833476913032043701676510023341657731843901} a^{6} + \frac{1316683071188964331561577060338982294015285511562487943408}{6294659757359515833476913032043701676510023341657731843901} a^{5} + \frac{2941555193420658032294406188361807994556453171432966465313}{6294659757359515833476913032043701676510023341657731843901} a^{4} + \frac{945374042274965479820374105914687835684104985921357601599}{6294659757359515833476913032043701676510023341657731843901} a^{3} + \frac{1082753694813366518618812776809604843840265369722569100988}{6294659757359515833476913032043701676510023341657731843901} a^{2} - \frac{1293410781670903720296819926649685509964278224518818944621}{6294659757359515833476913032043701676510023341657731843901} a + \frac{520104755560622648810030673681498899668162761375942934124}{6294659757359515833476913032043701676510023341657731843901}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}$, which has order $64$ (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:  $9$
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:  \( 2701772.21217 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ 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 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ 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.

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
17Data not computed
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$