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Properties

Label 16.0.13601600438...2304.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 13^{6}$
Root discriminant $18.13$
Ramified primes $2, 3, 13$
Class number $1$
Class group Trivial
Galois group $C_8:C_2^2$ (as 16T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, -112, 430, -960, 1406, -1512, 1370, -1108, 786, -520, 338, -168, 46, -8, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 8*x^13 + 46*x^12 - 168*x^11 + 338*x^10 - 520*x^9 + 786*x^8 - 1108*x^7 + 1370*x^6 - 1512*x^5 + 1406*x^4 - 960*x^3 + 430*x^2 - 112*x + 13)
 
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 - 8*x^13 + 46*x^12 - 168*x^11 + 338*x^10 - 520*x^9 + 786*x^8 - 1108*x^7 + 1370*x^6 - 1512*x^5 + 1406*x^4 - 960*x^3 + 430*x^2 - 112*x + 13, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 6 x^{14} - 8 x^{13} + 46 x^{12} - 168 x^{11} + 338 x^{10} - 520 x^{9} + 786 x^{8} - 1108 x^{7} + 1370 x^{6} - 1512 x^{5} + 1406 x^{4} - 960 x^{3} + 430 x^{2} - 112 x + 13 \)

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:  \(136016004388491362304=2^{32}\cdot 3^{8}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.13$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{26} a^{12} - \frac{1}{13} a^{10} + \frac{2}{13} a^{9} - \frac{3}{13} a^{8} - \frac{2}{13} a^{7} - \frac{3}{13} a^{6} - \frac{3}{13} a^{5} - \frac{1}{26} a^{4} + \frac{5}{13} a^{3} + \frac{4}{13} a^{2} - \frac{5}{13} a$, $\frac{1}{26} a^{13} - \frac{1}{13} a^{11} + \frac{2}{13} a^{10} - \frac{3}{13} a^{9} - \frac{2}{13} a^{8} - \frac{3}{13} a^{7} - \frac{3}{13} a^{6} - \frac{1}{26} a^{5} + \frac{5}{13} a^{4} + \frac{4}{13} a^{3} - \frac{5}{13} a^{2}$, $\frac{1}{442} a^{14} + \frac{5}{442} a^{13} + \frac{2}{221} a^{12} - \frac{71}{442} a^{11} - \frac{25}{221} a^{10} + \frac{8}{221} a^{9} - \frac{75}{442} a^{8} - \frac{30}{221} a^{7} + \frac{167}{442} a^{6} - \frac{83}{442} a^{5} - \frac{2}{17} a^{4} + \frac{3}{26} a^{3} + \frac{38}{221} a^{2} + \frac{100}{221} a - \frac{13}{34}$, $\frac{1}{114247718} a^{15} + \frac{30129}{114247718} a^{14} - \frac{418043}{114247718} a^{13} - \frac{1208057}{114247718} a^{12} + \frac{50447}{338011} a^{11} + \frac{24509239}{114247718} a^{10} + \frac{11682898}{57123859} a^{9} - \frac{7632826}{57123859} a^{8} - \frac{3683597}{114247718} a^{7} + \frac{17350117}{114247718} a^{6} - \frac{272327}{8788286} a^{5} - \frac{2278693}{6720454} a^{4} - \frac{17743814}{57123859} a^{3} - \frac{12025719}{114247718} a^{2} + \frac{5569872}{57123859} a + \frac{79877}{258479}$

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:  \( \frac{6639142}{57123859} a^{15} - \frac{7481177}{57123859} a^{14} - \frac{23148547}{57123859} a^{13} + \frac{28720955}{114247718} a^{12} + \frac{16199228}{4394143} a^{11} - \frac{315525233}{57123859} a^{10} - \frac{388192353}{57123859} a^{9} + \frac{1029439531}{57123859} a^{8} - \frac{1131794376}{57123859} a^{7} + \frac{2456307895}{57123859} a^{6} - \frac{320817685}{4394143} a^{5} + \frac{10499477289}{114247718} a^{4} - \frac{6680245606}{57123859} a^{3} + \frac{6827623869}{57123859} a^{2} - \frac{216874703}{3360227} a + \frac{63682999}{4394143} \) (order $12$)
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:  \( 25054.9537521 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T38):

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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.4.7488.1, 4.0.7488.1, \(\Q(\zeta_{12})\), 8.4.11662589952.1, 8.4.11662589952.2, 8.0.56070144.2

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$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}$