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Properties

Label 16.0.25554788874...0409.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 37^{10}$
Root discriminant $21.78$
Ramified primes $3, 37$
Class number $4$
Class group $[4]$
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![111, 666, 1221, 444, 85, 1084, -202, -540, 565, -514, 268, -166, 130, -78, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 78*x^13 + 130*x^12 - 166*x^11 + 268*x^10 - 514*x^9 + 565*x^8 - 540*x^7 - 202*x^6 + 1084*x^5 + 85*x^4 + 444*x^3 + 1221*x^2 + 666*x + 111)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 78*x^13 + 130*x^12 - 166*x^11 + 268*x^10 - 514*x^9 + 565*x^8 - 540*x^7 - 202*x^6 + 1084*x^5 + 85*x^4 + 444*x^3 + 1221*x^2 + 666*x + 111, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 78 x^{13} + 130 x^{12} - 166 x^{11} + 268 x^{10} - 514 x^{9} + 565 x^{8} - 540 x^{7} - 202 x^{6} + 1084 x^{5} + 85 x^{4} + 444 x^{3} + 1221 x^{2} + 666 x + 111 \)

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:  \(2555478887462114090409=3^{12}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.78$
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:  $3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{42} a^{13} + \frac{1}{42} a^{12} + \frac{4}{21} a^{11} + \frac{1}{21} a^{10} + \frac{3}{14} a^{9} + \frac{3}{14} a^{8} - \frac{13}{42} a^{7} + \frac{4}{21} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{2394} a^{14} + \frac{10}{1197} a^{13} - \frac{1}{42} a^{12} - \frac{17}{342} a^{11} + \frac{223}{1197} a^{10} - \frac{17}{798} a^{9} - \frac{299}{1197} a^{8} + \frac{311}{1197} a^{7} - \frac{185}{798} a^{6} - \frac{91}{342} a^{5} + \frac{169}{342} a^{4} - \frac{1}{42} a^{3} - \frac{134}{399} a^{2} + \frac{187}{798} a - \frac{16}{399}$, $\frac{1}{117689728620782178} a^{15} + \frac{11470158425615}{117689728620782178} a^{14} - \frac{208364600986285}{19614954770130363} a^{13} - \frac{1832888091759748}{58844864310391089} a^{12} + \frac{14212235601956465}{117689728620782178} a^{11} + \frac{604806680695531}{39229909540260726} a^{10} - \frac{8841487158538652}{58844864310391089} a^{9} - \frac{195982460543071}{3097098121599531} a^{8} + \frac{4048247900478802}{19614954770130363} a^{7} - \frac{6760324329936295}{117689728620782178} a^{6} + \frac{323101079608945}{884885177599866} a^{5} + \frac{1899759870058135}{19614954770130363} a^{4} - \frac{5782546456070977}{39229909540260726} a^{3} - \frac{3966689750411447}{39229909540260726} a^{2} + \frac{7086039828312223}{39229909540260726} a + \frac{1390313391683459}{6538318256710121}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( \frac{29307892456}{986651201529} a^{15} - \frac{82456597487}{328883733843} a^{14} + \frac{1048469024840}{986651201529} a^{13} - \frac{5520653047591}{1973302403058} a^{12} + \frac{3383118459211}{657767467686} a^{11} - \frac{14436526854047}{1973302403058} a^{10} + \frac{22484806508203}{1973302403058} a^{9} - \frac{6773284311190}{328883733843} a^{8} + \frac{52110837920273}{1973302403058} a^{7} - \frac{4033606006435}{140950171647} a^{6} + \frac{719033291315}{93966781098} a^{5} + \frac{27156090258826}{986651201529} a^{4} - \frac{6014111051743}{657767467686} a^{3} + \frac{3846996005777}{219255822562} a^{2} + \frac{9426403618505}{328883733843} a + \frac{785342047499}{93966781098} \) (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:  \( 34293.5127133 \)
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

\(\Q(\sqrt{-111}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{37}) \), 4.2.4107.1 x2, 4.0.333.1 x2, \(\Q(\sqrt{-3}, \sqrt{37})\), 8.0.151807041.1, 8.0.1366263369.1 x2, 8.0.50551744653.1 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
37Data not computed