↑

Properties

Label 16.0.51399544780...7056.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 7^{8}$
Root discriminant $17.06$
Ramified primes $2, 3, 7$
Class number $2$
Class group $[2]$
Galois group $C_2\wr C_2^2$ (as 16T149)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -72, 282, -738, 1411, -2060, 2414, -2322, 1858, -1288, 856, -550, 304, -126, 38, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 38*x^14 - 126*x^13 + 304*x^12 - 550*x^11 + 856*x^10 - 1288*x^9 + 1858*x^8 - 2322*x^7 + 2414*x^6 - 2060*x^5 + 1411*x^4 - 738*x^3 + 282*x^2 - 72*x + 9)
 
gp: K = bnfinit(x^16 - 8*x^15 + 38*x^14 - 126*x^13 + 304*x^12 - 550*x^11 + 856*x^10 - 1288*x^9 + 1858*x^8 - 2322*x^7 + 2414*x^6 - 2060*x^5 + 1411*x^4 - 738*x^3 + 282*x^2 - 72*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 38 x^{14} - 126 x^{13} + 304 x^{12} - 550 x^{11} + 856 x^{10} - 1288 x^{9} + 1858 x^{8} - 2322 x^{7} + 2414 x^{6} - 2060 x^{5} + 1411 x^{4} - 738 x^{3} + 282 x^{2} - 72 x + 9 \)

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:  \(51399544780206637056=2^{24}\cdot 3^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.06$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{2}{9} a^{7} + \frac{2}{9} a^{6} + \frac{1}{9} a^{5} - \frac{2}{9} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{284013} a^{14} - \frac{7}{284013} a^{13} - \frac{8143}{94671} a^{12} - \frac{42677}{284013} a^{11} + \frac{22286}{284013} a^{10} - \frac{11987}{94671} a^{9} - \frac{35047}{284013} a^{8} + \frac{76135}{284013} a^{7} - \frac{4321}{94671} a^{6} - \frac{98815}{284013} a^{5} - \frac{74600}{284013} a^{4} - \frac{35687}{94671} a^{3} + \frac{5389}{94671} a^{2} + \frac{3662}{31557} a + \frac{13762}{31557}$, $\frac{1}{852039} a^{15} - \frac{24478}{852039} a^{13} - \frac{24338}{852039} a^{12} + \frac{280}{31557} a^{11} + \frac{25370}{852039} a^{10} - \frac{2761}{852039} a^{9} + \frac{6716}{284013} a^{8} - \frac{332057}{852039} a^{7} + \frac{189128}{852039} a^{6} + \frac{41083}{94671} a^{5} - \frac{250577}{852039} a^{4} + \frac{39593}{284013} a^{3} + \frac{1198}{4239} a^{2} + \frac{13}{157} a + \frac{33220}{94671}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2771.45653108 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T149):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{3}) \), 4.2.1323.1, 4.2.21168.1, \(\Q(\sqrt{3}, \sqrt{7})\), 8.2.2389782528.1 x2, 8.4.448084224.2, 8.0.448084224.3 x2

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
Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$