↑

Properties

Label 16.0.15638154465...9601.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 61^{8}$
Root discriminant $28.16$
Ramified primes $13, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{8}$ (as 16T13)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, -78, 133, -64, 599, -802, 784, -227, 245, -283, 94, -186, 105, -29, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 29*x^13 + 105*x^12 - 186*x^11 + 94*x^10 - 283*x^9 + 245*x^8 - 227*x^7 + 784*x^6 - 802*x^5 + 599*x^4 - 64*x^3 + 133*x^2 - 78*x + 13)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 29*x^13 + 105*x^12 - 186*x^11 + 94*x^10 - 283*x^9 + 245*x^8 - 227*x^7 + 784*x^6 - 802*x^5 + 599*x^4 - 64*x^3 + 133*x^2 - 78*x + 13, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 29 x^{13} + 105 x^{12} - 186 x^{11} + 94 x^{10} - 283 x^{9} + 245 x^{8} - 227 x^{7} + 784 x^{6} - 802 x^{5} + 599 x^{4} - 64 x^{3} + 133 x^{2} - 78 x + 13 \)

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:  \(156381544652244701169601=13^{8}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.16$
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:  $13, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{65} a^{12} - \frac{11}{65} a^{10} + \frac{14}{65} a^{9} - \frac{3}{65} a^{8} + \frac{1}{5} a^{7} - \frac{7}{65} a^{6} + \frac{2}{13} a^{5} + \frac{23}{65} a^{4} + \frac{28}{65} a^{3} - \frac{16}{65} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{65} a^{13} - \frac{11}{65} a^{11} + \frac{14}{65} a^{10} - \frac{3}{65} a^{9} + \frac{1}{5} a^{8} - \frac{7}{65} a^{7} + \frac{2}{13} a^{6} + \frac{23}{65} a^{5} + \frac{28}{65} a^{4} - \frac{16}{65} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{65} a^{14} + \frac{14}{65} a^{11} + \frac{6}{65} a^{10} - \frac{28}{65} a^{9} + \frac{5}{13} a^{8} + \frac{23}{65} a^{7} + \frac{11}{65} a^{6} + \frac{8}{65} a^{5} - \frac{23}{65} a^{4} + \frac{9}{65} a^{3} + \frac{6}{65} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{2243034811951956425} a^{15} + \frac{9861306567528129}{2243034811951956425} a^{14} - \frac{9004437552154374}{2243034811951956425} a^{13} + \frac{12202414626572028}{2243034811951956425} a^{12} - \frac{11766417004927016}{118054463786945075} a^{11} - \frac{215112919520431}{4012584636765575} a^{10} - \frac{747691815985045604}{2243034811951956425} a^{9} - \frac{248007860420248671}{2243034811951956425} a^{8} + \frac{21485168664268997}{77346027998343325} a^{7} + \frac{743829905245355629}{2243034811951956425} a^{6} + \frac{477967806594660522}{2243034811951956425} a^{5} - \frac{440923165205962703}{2243034811951956425} a^{4} + \frac{916279379463885488}{2243034811951956425} a^{3} - \frac{51135540113327827}{118054463786945075} a^{2} + \frac{108543407964261}{9081112598995775} a - \frac{56453850119877433}{172541139380919725}$

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:  $7$
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:  \( 116747.923556 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{793}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{61})\), 4.4.48373.1 x2, 4.4.10309.1 x2, 8.8.395451064801.1, 8.0.30419312677.1 x4, 8.0.6482804341.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
$61$61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$