↑

Properties

Label 16.4.52703064995...1609.9
Degree $16$
Signature $[4, 6]$
Discriminant $17^{12}\cdot 67^{6}$
Root discriminant $40.51$
Ramified primes $17, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_4:C_4$ (as 16T26)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 40, -280, -618, 3045, 2210, -3484, -1084, 1586, 12, -518, 102, 60, 4, -18, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 18*x^14 + 4*x^13 + 60*x^12 + 102*x^11 - 518*x^10 + 12*x^9 + 1586*x^8 - 1084*x^7 - 3484*x^6 + 2210*x^5 + 3045*x^4 - 618*x^3 - 280*x^2 + 40*x + 16)
 
gp: K = bnfinit(x^16 - 2*x^15 - 18*x^14 + 4*x^13 + 60*x^12 + 102*x^11 - 518*x^10 + 12*x^9 + 1586*x^8 - 1084*x^7 - 3484*x^6 + 2210*x^5 + 3045*x^4 - 618*x^3 - 280*x^2 + 40*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 18 x^{14} + 4 x^{13} + 60 x^{12} + 102 x^{11} - 518 x^{10} + 12 x^{9} + 1586 x^{8} - 1084 x^{7} - 3484 x^{6} + 2210 x^{5} + 3045 x^{4} - 618 x^{3} - 280 x^{2} + 40 x + 16 \)

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:  \(52703064995487500398531609=17^{12}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.51$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} + \frac{3}{16} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{5}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{608} a^{13} - \frac{1}{304} a^{12} + \frac{7}{304} a^{11} - \frac{23}{608} a^{10} + \frac{1}{38} a^{9} - \frac{5}{76} a^{8} - \frac{21}{608} a^{7} + \frac{61}{304} a^{6} - \frac{55}{304} a^{5} - \frac{85}{608} a^{4} + \frac{13}{38} a^{3} - \frac{3}{19} a^{2} + \frac{31}{76} a - \frac{9}{19}$, $\frac{1}{1216} a^{14} - \frac{1}{1216} a^{13} - \frac{13}{608} a^{12} - \frac{85}{1216} a^{11} + \frac{69}{1216} a^{10} - \frac{31}{608} a^{9} + \frac{91}{1216} a^{8} + \frac{101}{1216} a^{7} + \frac{101}{608} a^{6} - \frac{119}{1216} a^{5} - \frac{257}{1216} a^{4} - \frac{1}{608} a^{3} - \frac{1}{4} a^{2} - \frac{5}{152} a + \frac{1}{76}$, $\frac{1}{109536783238061504} a^{15} + \frac{1211115221249}{6846048952378844} a^{14} + \frac{9523041486939}{109536783238061504} a^{13} - \frac{222440008591223}{109536783238061504} a^{12} - \frac{3354490226870501}{27384195809515376} a^{11} + \frac{2832072752749237}{109536783238061504} a^{10} - \frac{778079805051951}{109536783238061504} a^{9} + \frac{519475965880931}{6846048952378844} a^{8} + \frac{1176189154287419}{5765093854634816} a^{7} - \frac{13155354422991333}{109536783238061504} a^{6} - \frac{880820176256325}{27384195809515376} a^{5} + \frac{1741941320772775}{109536783238061504} a^{4} + \frac{11045228686305877}{54768391619030752} a^{3} - \frac{4650802986133025}{13692097904757688} a^{2} - \frac{2853657234214157}{13692097904757688} a + \frac{1695326070485343}{6846048952378844}$

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:  $9$
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:  \( 34224155.0112 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

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 $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.19363.1, 4.2.329171.1, 8.2.25120026523.1, 8.2.7259687665147.2, 8.4.108353547241.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$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}$