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Properties

Label 16.0.74796181841...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{12}\cdot 7^{8}$
Root discriminant $20.17$
Ramified primes $3, 5, 7$
Class number $2$
Class group $[2]$
Galois group $C_4 \times D_4$ (as 16T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -72, 212, -666, 1283, -1140, 401, -33, -122, 189, -56, -33, 38, -27, 13, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 13*x^14 - 27*x^13 + 38*x^12 - 33*x^11 - 56*x^10 + 189*x^9 - 122*x^8 - 33*x^7 + 401*x^6 - 1140*x^5 + 1283*x^4 - 666*x^3 + 212*x^2 - 72*x + 16)
 
gp: K = bnfinit(x^16 - 3*x^15 + 13*x^14 - 27*x^13 + 38*x^12 - 33*x^11 - 56*x^10 + 189*x^9 - 122*x^8 - 33*x^7 + 401*x^6 - 1140*x^5 + 1283*x^4 - 666*x^3 + 212*x^2 - 72*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 13 x^{14} - 27 x^{13} + 38 x^{12} - 33 x^{11} - 56 x^{10} + 189 x^{9} - 122 x^{8} - 33 x^{7} + 401 x^{6} - 1140 x^{5} + 1283 x^{4} - 666 x^{3} + 212 x^{2} - 72 x + 16 \)

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:  \(747961818418212890625=3^{12}\cdot 5^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.17$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{28} a^{14} + \frac{5}{28} a^{13} - \frac{5}{28} a^{12} - \frac{1}{4} a^{11} + \frac{3}{14} a^{10} + \frac{1}{28} a^{9} + \frac{5}{14} a^{8} + \frac{1}{28} a^{7} - \frac{2}{7} a^{6} - \frac{1}{28} a^{5} + \frac{3}{28} a^{4} + \frac{3}{14} a^{3} + \frac{9}{28} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{3472527646667576} a^{15} + \frac{58843951404083}{3472527646667576} a^{14} + \frac{74570777177673}{496075378095368} a^{13} - \frac{25096460486949}{3472527646667576} a^{12} - \frac{22250976339950}{434065955833447} a^{11} - \frac{39470334463627}{496075378095368} a^{10} - \frac{89519016368773}{1736263823333788} a^{9} - \frac{202733537841467}{3472527646667576} a^{8} - \frac{7347936774387}{124018844523842} a^{7} + \frac{447064853328399}{3472527646667576} a^{6} - \frac{1321375654588197}{3472527646667576} a^{5} + \frac{620291184879255}{1736263823333788} a^{4} + \frac{730753878793079}{3472527646667576} a^{3} - \frac{8245244365549}{434065955833447} a^{2} - \frac{158453254896142}{434065955833447} a - \frac{170184754204933}{434065955833447}$

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:  \( \frac{40165837097}{63836749208} a^{15} - \frac{101061112295}{63836749208} a^{14} + \frac{471648669765}{63836749208} a^{13} - \frac{853512625195}{63836749208} a^{12} + \frac{548165781515}{31918374604} a^{11} - \frac{774112506061}{63836749208} a^{10} - \frac{663523598565}{15959187302} a^{9} + \frac{6316814502665}{63836749208} a^{8} - \frac{866696337915}{31918374604} a^{7} - \frac{2329395373485}{63836749208} a^{6} + \frac{14941470194621}{63836749208} a^{5} - \frac{9632725890805}{15959187302} a^{4} + \frac{32290464627175}{63836749208} a^{3} - \frac{5028972336715}{31918374604} a^{2} + \frac{390884311990}{7979593651} a - \frac{149016562091}{7979593651} \) (order $6$)
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:  \( 9204.08316001 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_4$ (as 16T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.4.6125.1, 4.0.55125.1, 4.0.189.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.4725.1, 8.0.22325625.1, 8.0.558140625.1, 8.0.3038765625.1

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
$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}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$