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Properties

Label 16.0.10737418240...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{10}$
Root discriminant $15.47$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![5, 0, 24, -164, 292, -268, 216, -252, 377, -476, 444, -308, 174, -84, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 174*x^12 - 308*x^11 + 444*x^10 - 476*x^9 + 377*x^8 - 252*x^7 + 216*x^6 - 268*x^5 + 292*x^4 - 164*x^3 + 24*x^2 + 5)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 174*x^12 - 308*x^11 + 444*x^10 - 476*x^9 + 377*x^8 - 252*x^7 + 216*x^6 - 268*x^5 + 292*x^4 - 164*x^3 + 24*x^2 + 5, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 174 x^{12} - 308 x^{11} + 444 x^{10} - 476 x^{9} + 377 x^{8} - 252 x^{7} + 216 x^{6} - 268 x^{5} + 292 x^{4} - 164 x^{3} + 24 x^{2} + 5 \)

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:  \(10737418240000000000=2^{40}\cdot 5^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.47$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{12} - \frac{2}{25} a^{11} - \frac{1}{25} a^{10} - \frac{12}{25} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{3}{25} a^{6} - \frac{6}{25} a^{5} + \frac{8}{25} a^{4} + \frac{2}{5} a^{3} + \frac{9}{25} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{25} a^{13} + \frac{1}{25} a^{10} + \frac{6}{25} a^{9} + \frac{2}{5} a^{8} - \frac{12}{25} a^{7} - \frac{9}{25} a^{5} + \frac{11}{25} a^{4} - \frac{6}{25} a^{3} - \frac{12}{25} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{125} a^{14} + \frac{1}{125} a^{12} - \frac{1}{125} a^{11} + \frac{2}{25} a^{10} + \frac{53}{125} a^{9} + \frac{8}{125} a^{8} + \frac{4}{25} a^{7} + \frac{39}{125} a^{6} + \frac{8}{25} a^{5} - \frac{48}{125} a^{4} + \frac{58}{125} a^{3} - \frac{56}{125} a^{2} + \frac{3}{25} a - \frac{4}{25}$, $\frac{1}{714674375} a^{15} + \frac{2162561}{714674375} a^{14} + \frac{1204116}{714674375} a^{13} - \frac{723766}{142934875} a^{12} + \frac{701904}{714674375} a^{11} - \frac{52541432}{714674375} a^{10} - \frac{251330489}{714674375} a^{9} + \frac{200427158}{714674375} a^{8} - \frac{157388971}{714674375} a^{7} + \frac{304945999}{714674375} a^{6} + \frac{269906022}{714674375} a^{5} - \frac{508181}{1143479} a^{4} + \frac{93686167}{714674375} a^{3} + \frac{333653609}{714674375} a^{2} + \frac{53336659}{142934875} a + \frac{49238221}{142934875}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{19514364}{142934875} a^{15} - \frac{135108602}{142934875} a^{14} + \frac{470951394}{142934875} a^{13} - \frac{1074844866}{142934875} a^{12} + \frac{2029740142}{142934875} a^{11} - \frac{3321816998}{142934875} a^{10} + \frac{4089922786}{142934875} a^{9} - \frac{3184856161}{142934875} a^{8} + \frac{1658339946}{142934875} a^{7} - \frac{981223868}{142934875} a^{6} + \frac{1622332478}{142934875} a^{5} - \frac{2503234287}{142934875} a^{4} + \frac{415054114}{28586975} a^{3} + \frac{208773762}{142934875} a^{2} - \frac{98231972}{28586975} a - \frac{22567882}{28586975} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3713.68913152 \) (assuming GRH)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), 4.0.1280.1 x2, 4.2.1600.1 x2, \(\Q(i, \sqrt{5})\), 8.0.204800000.3 x2, 8.0.40960000.2, 8.0.655360000.3 x2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$