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Properties

Label 16.0.18518907289...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 41^{4}$
Root discriminant $16.00$
Ramified primes $2, 5, 41$
Class number $2$
Class group $[2]$
Galois group $C_2\wr C_2^2$ (as 16T149)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 9, -36, 61, -88, 185, -220, 276, -300, 235, -156, 91, -38, 16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 16*x^14 - 38*x^13 + 91*x^12 - 156*x^11 + 235*x^10 - 300*x^9 + 276*x^8 - 220*x^7 + 185*x^6 - 88*x^5 + 61*x^4 - 36*x^3 + 9*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 16*x^14 - 38*x^13 + 91*x^12 - 156*x^11 + 235*x^10 - 300*x^9 + 276*x^8 - 220*x^7 + 185*x^6 - 88*x^5 + 61*x^4 - 36*x^3 + 9*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 16 x^{14} - 38 x^{13} + 91 x^{12} - 156 x^{11} + 235 x^{10} - 300 x^{9} + 276 x^{8} - 220 x^{7} + 185 x^{6} - 88 x^{5} + 61 x^{4} - 36 x^{3} + 9 x^{2} - 2 x + 1 \)

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:  \(18518907289600000000=2^{24}\cdot 5^{8}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.00$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{20} a^{12} + \frac{3}{10} a^{11} + \frac{7}{20} a^{10} + \frac{2}{5} a^{9} - \frac{1}{20} a^{8} - \frac{1}{10} a^{7} + \frac{1}{20} a^{6} + \frac{3}{10} a^{5} - \frac{1}{5} a^{4} - \frac{3}{10} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{9}{20}$, $\frac{1}{60} a^{13} - \frac{1}{60} a^{12} - \frac{1}{4} a^{11} - \frac{7}{20} a^{10} + \frac{1}{20} a^{9} - \frac{1}{4} a^{8} - \frac{1}{12} a^{7} - \frac{7}{20} a^{6} + \frac{7}{30} a^{5} - \frac{3}{10} a^{4} - \frac{1}{10} a^{3} + \frac{7}{15} a^{2} + \frac{5}{12} a + \frac{17}{60}$, $\frac{1}{60} a^{14} - \frac{1}{60} a^{12} - \frac{1}{10} a^{11} + \frac{9}{20} a^{10} - \frac{1}{5} a^{9} + \frac{5}{12} a^{8} + \frac{1}{15} a^{7} + \frac{2}{15} a^{6} + \frac{13}{30} a^{5} - \frac{2}{5} a^{4} - \frac{2}{15} a^{3} - \frac{7}{60} a^{2} - \frac{3}{10} a - \frac{7}{15}$, $\frac{1}{356637780} a^{15} - \frac{288191}{35663778} a^{14} + \frac{40397}{11887926} a^{13} - \frac{3425291}{356637780} a^{12} - \frac{4872189}{19813210} a^{11} - \frac{49216447}{118879260} a^{10} - \frac{28594987}{178318890} a^{9} - \frac{79969387}{356637780} a^{8} - \frac{64209059}{356637780} a^{7} - \frac{136571489}{356637780} a^{6} - \frac{78984097}{178318890} a^{5} + \frac{15924715}{35663778} a^{4} - \frac{30403103}{118879260} a^{3} - \frac{639833}{9906605} a^{2} + \frac{56056553}{118879260} a - \frac{107678969}{356637780}$

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:  \( 613.314800422 \)
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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.0.65600.5, \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.65600.2, 8.0.4303360000.6 x2, 8.0.4303360000.3, 8.4.104960000.1 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.4.2.1$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}$
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}$
$41$41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$