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Properties

Label 16.0.76029470982...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{10}\cdot 29^{4}$
Root discriminant $31.09$
Ramified primes $2, 3, 5, 29$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T210)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![529, -1840, 3748, -3764, 1459, 1024, -724, -784, 1002, -412, 116, -8, -32, 16, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 16*x^13 - 32*x^12 - 8*x^11 + 116*x^10 - 412*x^9 + 1002*x^8 - 784*x^7 - 724*x^6 + 1024*x^5 + 1459*x^4 - 3764*x^3 + 3748*x^2 - 1840*x + 529)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 + 16*x^13 - 32*x^12 - 8*x^11 + 116*x^10 - 412*x^9 + 1002*x^8 - 784*x^7 - 724*x^6 + 1024*x^5 + 1459*x^4 - 3764*x^3 + 3748*x^2 - 1840*x + 529, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} + 16 x^{13} - 32 x^{12} - 8 x^{11} + 116 x^{10} - 412 x^{9} + 1002 x^{8} - 784 x^{7} - 724 x^{6} + 1024 x^{5} + 1459 x^{4} - 3764 x^{3} + 3748 x^{2} - 1840 x + 529 \)

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:  \(760294709821440000000000=2^{24}\cdot 3^{8}\cdot 5^{10}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.09$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{3}{10} a^{2} - \frac{1}{5} a - \frac{1}{10}$, $\frac{1}{10} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{10} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a - \frac{1}{5}$, $\frac{1}{10} a^{10} - \frac{1}{5} a^{6} - \frac{1}{10} a^{4} - \frac{2}{5} a^{3} - \frac{1}{10} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{50} a^{11} + \frac{1}{50} a^{10} - \frac{1}{50} a^{9} + \frac{1}{50} a^{8} + \frac{4}{25} a^{7} + \frac{3}{25} a^{6} - \frac{1}{2} a^{5} + \frac{19}{50} a^{4} + \frac{3}{25} a^{3} - \frac{11}{25} a^{2} - \frac{3}{10} a + \frac{19}{50}$, $\frac{1}{50} a^{12} - \frac{1}{25} a^{10} + \frac{1}{25} a^{9} + \frac{1}{25} a^{8} + \frac{4}{25} a^{7} + \frac{9}{50} a^{6} - \frac{3}{25} a^{5} + \frac{17}{50} a^{4} + \frac{6}{25} a^{3} + \frac{11}{25} a^{2} - \frac{3}{25} a - \frac{7}{25}$, $\frac{1}{11950} a^{13} - \frac{67}{11950} a^{12} + \frac{1}{2390} a^{11} + \frac{219}{5975} a^{10} + \frac{43}{5975} a^{9} + \frac{461}{11950} a^{8} + \frac{5169}{11950} a^{7} - \frac{4297}{11950} a^{6} + \frac{2047}{5975} a^{5} + \frac{5481}{11950} a^{4} - \frac{87}{2390} a^{3} - \frac{319}{11950} a^{2} + \frac{1889}{5975} a - \frac{3349}{11950}$, $\frac{1}{657250} a^{14} + \frac{2}{328625} a^{13} - \frac{329}{131450} a^{12} + \frac{516}{328625} a^{11} - \frac{3528}{328625} a^{10} + \frac{11051}{328625} a^{9} + \frac{14409}{328625} a^{8} - \frac{16023}{65725} a^{7} + \frac{133077}{328625} a^{6} + \frac{136964}{328625} a^{5} - \frac{217149}{657250} a^{4} - \frac{151593}{328625} a^{3} - \frac{40867}{131450} a^{2} - \frac{160689}{328625} a + \frac{106859}{657250}$, $\frac{1}{2191928750} a^{15} - \frac{1}{43838575} a^{14} + \frac{45659}{2191928750} a^{13} + \frac{8574661}{1095964375} a^{12} + \frac{4898478}{1095964375} a^{11} - \frac{91343749}{2191928750} a^{10} + \frac{6415144}{219192875} a^{9} - \frac{736321}{1095964375} a^{8} - \frac{29637138}{1095964375} a^{7} - \frac{547847144}{1095964375} a^{6} + \frac{1056966369}{2191928750} a^{5} - \frac{137506827}{438385750} a^{4} + \frac{645148699}{2191928750} a^{3} - \frac{573742973}{2191928750} a^{2} - \frac{37777869}{2191928750} a + \frac{1706614}{47650625}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 58897.8835353 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T210):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{30}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{5}, \sqrt{6})\), 4.0.6525.1, 4.0.46400.1, 8.4.30067200000.1, 8.4.30067200000.3, 8.0.174389760000.15

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

Sibling fields

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$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}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$