↑

Properties

Label 16.4.29129459172...6304.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{32}\cdot 7^{14}$
Root discriminant $21.95$
Ramified primes $2, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_8:C_2^2$ (as 16T45)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, -64, 16, 112, -196, 112, 224, -540, 345, 136, -238, 56, 14, 0, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 14*x^12 + 56*x^11 - 238*x^10 + 136*x^9 + 345*x^8 - 540*x^7 + 224*x^6 + 112*x^5 - 196*x^4 + 112*x^3 + 16*x^2 - 64*x + 32)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 14*x^12 + 56*x^11 - 238*x^10 + 136*x^9 + 345*x^8 - 540*x^7 + 224*x^6 + 112*x^5 - 196*x^4 + 112*x^3 + 16*x^2 - 64*x + 32, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} + 14 x^{12} + 56 x^{11} - 238 x^{10} + 136 x^{9} + 345 x^{8} - 540 x^{7} + 224 x^{6} + 112 x^{5} - 196 x^{4} + 112 x^{3} + 16 x^{2} - 64 x + 32 \)

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:  \(2912945917279080546304=2^{32}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.95$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{112} a^{12} + \frac{3}{56} a^{11} - \frac{1}{16} a^{10} + \frac{1}{7} a^{9} - \frac{5}{112} a^{8} - \frac{13}{56} a^{7} - \frac{1}{16} a^{6} + \frac{1}{7} a^{5} + \frac{13}{56} a^{4} - \frac{1}{2} a^{3} + \frac{2}{7} a^{2} + \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{112} a^{13} - \frac{1}{112} a^{11} + \frac{1}{56} a^{10} - \frac{3}{112} a^{9} - \frac{3}{14} a^{8} - \frac{5}{112} a^{7} + \frac{1}{56} a^{6} + \frac{1}{4} a^{5} - \frac{1}{7} a^{4} - \frac{13}{28} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{224} a^{14} + \frac{1}{28} a^{11} + \frac{9}{112} a^{10} + \frac{3}{14} a^{9} + \frac{9}{112} a^{8} - \frac{3}{28} a^{7} + \frac{7}{32} a^{6} + \frac{1}{4} a^{5} + \frac{1}{112} a^{4} - \frac{13}{28} a^{3} - \frac{1}{14} a^{2} + \frac{2}{7} a - \frac{5}{14}$, $\frac{1}{5586112} a^{15} + \frac{1545}{698264} a^{14} + \frac{5979}{1396528} a^{13} - \frac{4345}{1396528} a^{12} - \frac{277}{75488} a^{11} - \frac{129005}{1396528} a^{10} + \frac{407271}{2793056} a^{9} + \frac{71221}{1396528} a^{8} - \frac{97877}{798016} a^{7} - \frac{341549}{1396528} a^{6} + \frac{1337897}{2793056} a^{5} - \frac{83029}{349132} a^{4} - \frac{169247}{349132} a^{3} - \frac{39870}{87283} a^{2} - \frac{87749}{349132} a - \frac{18941}{87283}$

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:  \( 342362.827423 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{2}) \), 4.2.21952.1, 4.2.87808.2, \(\Q(\sqrt{2}, \sqrt{7})\), 8.4.7710244864.2

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 8 siblings: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
$7$7.8.7.2$x^{8} - 7$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$
7.8.7.2$x^{8} - 7$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$