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Properties

Label 16.0.20106256000...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 19^{2}\cdot 59^{2}$
Root discriminant $16.09$
Ramified primes $2, 5, 19, 59$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2\wr C_4$ (as 16T261)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -1024, 1728, -1472, 448, 288, -264, -56, 153, -28, -66, 36, 28, -46, 27, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 27*x^14 - 46*x^13 + 28*x^12 + 36*x^11 - 66*x^10 - 28*x^9 + 153*x^8 - 56*x^7 - 264*x^6 + 288*x^5 + 448*x^4 - 1472*x^3 + 1728*x^2 - 1024*x + 256)
 
gp: K = bnfinit(x^16 - 8*x^15 + 27*x^14 - 46*x^13 + 28*x^12 + 36*x^11 - 66*x^10 - 28*x^9 + 153*x^8 - 56*x^7 - 264*x^6 + 288*x^5 + 448*x^4 - 1472*x^3 + 1728*x^2 - 1024*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 27 x^{14} - 46 x^{13} + 28 x^{12} + 36 x^{11} - 66 x^{10} - 28 x^{9} + 153 x^{8} - 56 x^{7} - 264 x^{6} + 288 x^{5} + 448 x^{4} - 1472 x^{3} + 1728 x^{2} - 1024 x + 256 \)

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:  \(20106256000000000000=2^{16}\cdot 5^{12}\cdot 19^{2}\cdot 59^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.09$
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:  $2, 5, 19, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} + \frac{1}{8} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{16} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{11} + \frac{1}{16} a^{10} - \frac{1}{4} a^{9} - \frac{3}{8} a^{8} - \frac{1}{16} a^{7} + \frac{1}{8} a^{6} - \frac{15}{32} a^{5} - \frac{1}{2} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{12} + \frac{1}{32} a^{11} - \frac{1}{8} a^{10} - \frac{3}{16} a^{9} - \frac{1}{32} a^{8} + \frac{1}{16} a^{7} + \frac{17}{64} a^{6} - \frac{1}{4} a^{5} + \frac{7}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{128} a^{15} - \frac{1}{128} a^{13} + \frac{1}{64} a^{12} - \frac{1}{16} a^{11} - \frac{3}{32} a^{10} - \frac{1}{64} a^{9} - \frac{15}{32} a^{8} - \frac{47}{128} a^{7} + \frac{3}{8} a^{6} + \frac{7}{32} a^{5} + \frac{7}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$

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:  \( -\frac{413}{64} a^{15} + \frac{2739}{64} a^{14} - \frac{7397}{64} a^{13} + \frac{8847}{64} a^{12} + \frac{37}{4} a^{11} - \frac{3519}{16} a^{10} + \frac{3967}{32} a^{9} + \frac{11247}{32} a^{8} - \frac{32333}{64} a^{7} - \frac{21325}{64} a^{6} + \frac{39929}{32} a^{5} - \frac{1157}{8} a^{4} - \frac{12375}{4} a^{3} + 5255 a^{2} - 3935 a + 1202 \) (order $20$)
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:  \( 7134.1584325 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2\wr C_4$ (as 16T261):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2\times C_2\wr C_4$
Character table for $C_2\times C_2\wr C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{5})\), \(\Q(i, \sqrt{5})\), 8.0.4484000000.1, 8.8.4484000000.1, \(\Q(\zeta_{20})\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$