↑

Properties

Label 16.0.14215874243...8544.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{14}\cdot 11^{6}$
Root discriminant $18.18$
Ramified primes $2, 3, 11$
Class number $2$
Class group $[2]$
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![169, -1092, 3086, -4722, 3634, -78, -2494, 1968, -89, -726, 374, 48, -110, 30, 8, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 8*x^14 + 30*x^13 - 110*x^12 + 48*x^11 + 374*x^10 - 726*x^9 - 89*x^8 + 1968*x^7 - 2494*x^6 - 78*x^5 + 3634*x^4 - 4722*x^3 + 3086*x^2 - 1092*x + 169)
 
gp: K = bnfinit(x^16 - 6*x^15 + 8*x^14 + 30*x^13 - 110*x^12 + 48*x^11 + 374*x^10 - 726*x^9 - 89*x^8 + 1968*x^7 - 2494*x^6 - 78*x^5 + 3634*x^4 - 4722*x^3 + 3086*x^2 - 1092*x + 169, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 8 x^{14} + 30 x^{13} - 110 x^{12} + 48 x^{11} + 374 x^{10} - 726 x^{9} - 89 x^{8} + 1968 x^{7} - 2494 x^{6} - 78 x^{5} + 3634 x^{4} - 4722 x^{3} + 3086 x^{2} - 1092 x + 169 \)

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:  \(142158742435915628544=2^{24}\cdot 3^{14}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.18$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{1015415717966} a^{15} + \frac{10610043051}{507707858983} a^{14} - \frac{113089390800}{507707858983} a^{13} + \frac{52958216586}{507707858983} a^{12} - \frac{15431813247}{1015415717966} a^{11} - \frac{1134664575}{10468203278} a^{10} + \frac{573485650}{3114772141} a^{9} - \frac{33700977217}{1015415717966} a^{8} + \frac{354577802477}{1015415717966} a^{7} + \frac{179408707951}{1015415717966} a^{6} + \frac{57433014185}{507707858983} a^{5} - \frac{25848517807}{78108901382} a^{4} - \frac{203422065928}{507707858983} a^{3} + \frac{1768012083}{10468203278} a^{2} + \frac{241920168135}{507707858983} a + \frac{1268148573}{78108901382}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{51302411831}{507707858983} a^{15} + \frac{772403962777}{1015415717966} a^{14} - \frac{703891736898}{507707858983} a^{13} - \frac{1756562999769}{507707858983} a^{12} + \frac{8299136040178}{507707858983} a^{11} - \frac{104586679741}{10468203278} a^{10} - \frac{168818132786}{3114772141} a^{9} + \frac{115315598051661}{1015415717966} a^{8} + \frac{7078462683315}{507707858983} a^{7} - \frac{316204537175659}{1015415717966} a^{6} + \frac{181698271553146}{507707858983} a^{5} + \frac{7033524376985}{78108901382} a^{4} - \frac{295164147367717}{507707858983} a^{3} + \frac{3144512531632}{5234101639} a^{2} - \frac{146746796246337}{507707858983} a + \frac{4556371711235}{78108901382} \) (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:  \( 12824.3091244 \)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.4.4752.1, 4.0.4752.1, \(\Q(\zeta_{12})\), 8.0.22581504.2

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
3Data not computed
$11$11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$