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Properties

Label 16.0.53084160000...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 3^{4}\cdot 5^{12}$
Root discriminant $14.80$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 8, 16, 8, -16, 0, -20, -14, 20, 0, 16, 8, -16, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 16*x^13 + 8*x^12 + 16*x^11 + 20*x^9 - 14*x^8 - 20*x^7 - 16*x^5 + 8*x^4 + 16*x^3 + 8*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 16*x^13 + 8*x^12 + 16*x^11 + 20*x^9 - 14*x^8 - 20*x^7 - 16*x^5 + 8*x^4 + 16*x^3 + 8*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{14} - 16 x^{13} + 8 x^{12} + 16 x^{11} + 20 x^{9} - 14 x^{8} - 20 x^{7} - 16 x^{5} + 8 x^{4} + 16 x^{3} + 8 x^{2} + 4 x + 1 \)

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:  \(5308416000000000000=2^{28}\cdot 3^{4}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.80$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{60} a^{12} + \frac{1}{20} a^{11} - \frac{1}{12} a^{9} - \frac{1}{15} a^{8} + \frac{1}{6} a^{7} - \frac{1}{15} a^{6} - \frac{1}{6} a^{5} - \frac{19}{60} a^{4} + \frac{1}{12} a^{3} - \frac{1}{20} a + \frac{4}{15}$, $\frac{1}{60} a^{13} + \frac{1}{10} a^{11} - \frac{1}{12} a^{10} - \frac{1}{15} a^{9} + \frac{7}{60} a^{8} - \frac{1}{15} a^{7} + \frac{1}{30} a^{6} - \frac{19}{60} a^{5} + \frac{1}{30} a^{4} - \frac{1}{2} a^{3} - \frac{1}{20} a^{2} - \frac{1}{3} a - \frac{1}{20}$, $\frac{1}{120} a^{14} - \frac{1}{120} a^{12} + \frac{1}{30} a^{11} + \frac{11}{120} a^{10} + \frac{1}{10} a^{9} + \frac{3}{40} a^{8} + \frac{13}{30} a^{7} + \frac{3}{40} a^{6} - \frac{2}{5} a^{5} - \frac{17}{120} a^{4} - \frac{1}{15} a^{3} - \frac{1}{24} a^{2} - \frac{1}{10} a - \frac{7}{120}$, $\frac{1}{600} a^{15} - \frac{1}{600} a^{14} - \frac{1}{120} a^{13} - \frac{1}{600} a^{12} + \frac{1}{24} a^{11} - \frac{13}{200} a^{10} + \frac{73}{600} a^{9} - \frac{17}{200} a^{8} + \frac{71}{200} a^{7} - \frac{221}{600} a^{6} - \frac{133}{600} a^{5} + \frac{59}{120} a^{4} - \frac{89}{200} a^{3} + \frac{49}{120} a^{2} - \frac{47}{600} a + \frac{253}{600}$

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:  \( \frac{211}{100} a^{15} - \frac{463}{50} a^{14} + \frac{411}{20} a^{13} - \frac{2103}{50} a^{12} + \frac{679}{20} a^{11} + \frac{953}{50} a^{10} - \frac{617}{100} a^{9} + \frac{2257}{50} a^{8} - \frac{4777}{100} a^{7} - \frac{1053}{50} a^{6} + \frac{667}{100} a^{5} - \frac{369}{10} a^{4} + \frac{3183}{100} a^{3} + \frac{199}{10} a^{2} + \frac{1043}{100} a + \frac{249}{50} \) (order $4$)
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:  \( 1070.46014902 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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 $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), 4.0.8000.1 x2, \(\Q(i, \sqrt{5})\), 4.2.2000.1 x2, 8.0.2304000000.2 x2, 8.0.115200000.1 x2, 8.0.64000000.3

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$