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Properties

Label 16.0.54036008766...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 19^{12}$
Root discriminant $30.43$
Ramified primes $5, 19$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T157)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7031, -31190, 67089, -89994, 82214, -52632, 22924, -5566, -449, 1124, -594, 199, -24, -26, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 21*x^14 - 26*x^13 - 24*x^12 + 199*x^11 - 594*x^10 + 1124*x^9 - 449*x^8 - 5566*x^7 + 22924*x^6 - 52632*x^5 + 82214*x^4 - 89994*x^3 + 67089*x^2 - 31190*x + 7031)
 
gp: K = bnfinit(x^16 - 7*x^15 + 21*x^14 - 26*x^13 - 24*x^12 + 199*x^11 - 594*x^10 + 1124*x^9 - 449*x^8 - 5566*x^7 + 22924*x^6 - 52632*x^5 + 82214*x^4 - 89994*x^3 + 67089*x^2 - 31190*x + 7031, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 21 x^{14} - 26 x^{13} - 24 x^{12} + 199 x^{11} - 594 x^{10} + 1124 x^{9} - 449 x^{8} - 5566 x^{7} + 22924 x^{6} - 52632 x^{5} + 82214 x^{4} - 89994 x^{3} + 67089 x^{2} - 31190 x + 7031 \)

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:  \(540360087662636962890625=5^{12}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.43$
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:  $5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{25} a^{10} + \frac{1}{25} a^{9} - \frac{1}{25} a^{8} + \frac{6}{25} a^{7} - \frac{11}{25} a^{6} + \frac{4}{25} a^{5} + \frac{7}{25} a^{4} + \frac{8}{25} a^{3} + \frac{7}{25} a^{2} + \frac{3}{25} a - \frac{1}{25}$, $\frac{1}{25} a^{11} - \frac{2}{25} a^{9} + \frac{2}{25} a^{8} - \frac{2}{25} a^{7} + \frac{1}{5} a^{6} + \frac{3}{25} a^{5} - \frac{4}{25} a^{4} - \frac{1}{25} a^{3} + \frac{6}{25} a^{2} - \frac{9}{25} a - \frac{4}{25}$, $\frac{1}{25} a^{12} - \frac{1}{25} a^{9} + \frac{1}{25} a^{8} + \frac{12}{25} a^{7} + \frac{11}{25} a^{6} - \frac{1}{25} a^{5} + \frac{3}{25} a^{4} + \frac{7}{25} a^{3} - \frac{1}{5} a^{2} + \frac{12}{25} a - \frac{12}{25}$, $\frac{1}{25} a^{13} + \frac{2}{25} a^{9} + \frac{1}{25} a^{8} - \frac{3}{25} a^{7} - \frac{7}{25} a^{6} + \frac{7}{25} a^{5} + \frac{4}{25} a^{4} + \frac{3}{25} a^{3} - \frac{11}{25} a^{2} + \frac{6}{25} a - \frac{11}{25}$, $\frac{1}{125} a^{14} - \frac{1}{125} a^{13} + \frac{1}{125} a^{12} - \frac{1}{125} a^{11} + \frac{1}{125} a^{10} + \frac{4}{125} a^{9} + \frac{11}{125} a^{8} + \frac{49}{125} a^{7} + \frac{41}{125} a^{6} - \frac{56}{125} a^{5} + \frac{54}{125} a^{4} + \frac{1}{125} a^{3} - \frac{6}{125} a^{2} + \frac{36}{125} a - \frac{41}{125}$, $\frac{1}{173332126211875} a^{15} + \frac{247318963651}{173332126211875} a^{14} + \frac{1249043686704}{173332126211875} a^{13} + \frac{2399347329181}{173332126211875} a^{12} + \frac{1150926712074}{173332126211875} a^{11} - \frac{534767409759}{173332126211875} a^{10} + \frac{16764296781434}{173332126211875} a^{9} + \frac{10823259194321}{173332126211875} a^{8} - \frac{26685762523206}{173332126211875} a^{7} - \frac{74423468665589}{173332126211875} a^{6} - \frac{21947982932238}{173332126211875} a^{5} - \frac{85874471823861}{173332126211875} a^{4} + \frac{31824919368251}{173332126211875} a^{3} + \frac{2888560900666}{5976969869375} a^{2} + \frac{48871815072251}{173332126211875} a + \frac{33606882590618}{173332126211875}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( -1 \) (order $2$)
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:  \( 127070.564298 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T157):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.45125.1, 8.2.38689046875.2 x2, 8.4.735091890625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
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}$
$19$19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$