↑

Properties

Label 16.0.18830918429...0096.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 3^{12}\cdot 7^{12}$
Root discriminant $13.87$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $C_2^2:D_4$ (as 16T43)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 63, 108, -153, -330, 282, 297, -384, 52, 187, -176, 34, 55, -56, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 55*x^12 + 34*x^11 - 176*x^10 + 187*x^9 + 52*x^8 - 384*x^7 + 297*x^6 + 282*x^5 - 330*x^4 - 153*x^3 + 108*x^2 + 63*x + 9)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 55*x^12 + 34*x^11 - 176*x^10 + 187*x^9 + 52*x^8 - 384*x^7 + 297*x^6 + 282*x^5 - 330*x^4 - 153*x^3 + 108*x^2 + 63*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 55 x^{12} + 34 x^{11} - 176 x^{10} + 187 x^{9} + 52 x^{8} - 384 x^{7} + 297 x^{6} + 282 x^{5} - 330 x^{4} - 153 x^{3} + 108 x^{2} + 63 x + 9 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1883091842914980096=2^{8}\cdot 3^{12}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.87$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{4}$, $\frac{1}{603} a^{14} - \frac{7}{603} a^{13} + \frac{25}{201} a^{12} + \frac{43}{603} a^{11} - \frac{73}{603} a^{10} + \frac{8}{67} a^{9} - \frac{26}{603} a^{8} - \frac{172}{603} a^{7} + \frac{28}{201} a^{6} + \frac{29}{67} a^{5} + \frac{30}{67} a^{4} + \frac{58}{201} a^{3} + \frac{17}{67} a^{2} - \frac{28}{67} a + \frac{31}{67}$, $\frac{1}{959373} a^{15} + \frac{788}{959373} a^{14} + \frac{34082}{319791} a^{13} - \frac{1436}{959373} a^{12} + \frac{93206}{959373} a^{11} + \frac{43123}{319791} a^{10} - \frac{42884}{959373} a^{9} + \frac{15338}{959373} a^{8} + \frac{92200}{319791} a^{7} - \frac{119492}{319791} a^{6} + \frac{28378}{106597} a^{5} + \frac{3518}{8643} a^{4} + \frac{590}{2479} a^{3} + \frac{23805}{106597} a^{2} + \frac{3164}{106597} a - \frac{49323}{106597}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1}{67} a^{14} + \frac{7}{67} a^{13} - \frac{91}{201} a^{12} + \frac{91}{67} a^{11} - \frac{518}{201} a^{10} + \frac{196}{67} a^{9} - \frac{56}{201} a^{8} - \frac{1159}{201} a^{7} + \frac{1624}{201} a^{6} - \frac{448}{201} a^{5} - \frac{672}{67} a^{4} + \frac{1099}{67} a^{3} + \frac{182}{67} a^{2} - \frac{686}{67} a - \frac{145}{67} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 805.821091977 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T43):

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 $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), 4.0.189.1 x2, 4.2.1323.1 x2, 4.0.12348.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 4.0.1372.1, 8.0.152473104.1, 8.0.1372257936.1, 8.0.1750329.1

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 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$7$7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$