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Properties

Label 16.0.95224002328...6336.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{20}\cdot 3^{8}\cdot 7^{12}$
Root discriminant $17.73$
Ramified primes $2, 3, 7$
Class number $2$
Class group $[2]$
Galois group $C_2^2\wr C_2$ (as 16T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 16, 30, 29, -8, -42, -10, 28, 10, -42, 8, 29, -30, 16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 16*x^14 - 30*x^13 + 29*x^12 + 8*x^11 - 42*x^10 + 10*x^9 + 28*x^8 - 10*x^7 - 42*x^6 - 8*x^5 + 29*x^4 + 30*x^3 + 16*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 16*x^14 - 30*x^13 + 29*x^12 + 8*x^11 - 42*x^10 + 10*x^9 + 28*x^8 - 10*x^7 - 42*x^6 - 8*x^5 + 29*x^4 + 30*x^3 + 16*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 16 x^{14} - 30 x^{13} + 29 x^{12} + 8 x^{11} - 42 x^{10} + 10 x^{9} + 28 x^{8} - 10 x^{7} - 42 x^{6} - 8 x^{5} + 29 x^{4} + 30 x^{3} + 16 x^{2} + 4 x + 1 \)

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:  \(95224002328145166336=2^{20}\cdot 3^{8}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.73$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{222} a^{13} - \frac{8}{111} a^{12} - \frac{8}{111} a^{11} + \frac{1}{74} a^{10} + \frac{5}{111} a^{9} + \frac{5}{74} a^{8} + \frac{55}{111} a^{7} - \frac{34}{111} a^{6} + \frac{8}{111} a^{5} - \frac{23}{222} a^{4} + \frac{46}{111} a^{3} - \frac{89}{222} a^{2} + \frac{5}{74} a - \frac{40}{111}$, $\frac{1}{222} a^{14} - \frac{13}{222} a^{12} + \frac{1}{37} a^{11} - \frac{8}{111} a^{10} - \frac{5}{111} a^{9} + \frac{17}{222} a^{8} - \frac{5}{111} a^{7} + \frac{25}{74} a^{6} - \frac{13}{111} a^{5} - \frac{9}{37} a^{4} + \frac{7}{111} a^{3} - \frac{20}{111} a^{2} + \frac{43}{111} a - \frac{16}{37}$, $\frac{1}{222} a^{15} - \frac{17}{222} a^{12} - \frac{1}{111} a^{11} - \frac{4}{111} a^{10} - \frac{1}{222} a^{9} + \frac{25}{222} a^{7} - \frac{16}{37} a^{6} + \frac{1}{37} a^{5} - \frac{50}{111} a^{4} + \frac{23}{111} a^{3} - \frac{12}{37} a^{2} + \frac{33}{74} a + \frac{107}{222}$

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:  \( \frac{44}{111} a^{15} - \frac{341}{222} a^{14} + \frac{664}{111} a^{13} - \frac{1145}{111} a^{12} + \frac{1505}{222} a^{11} + \frac{1334}{111} a^{10} - \frac{2947}{111} a^{9} + \frac{827}{111} a^{8} + \frac{3653}{222} a^{7} - \frac{1132}{111} a^{6} - \frac{3731}{222} a^{5} - \frac{68}{111} a^{4} + \frac{1451}{111} a^{3} + \frac{2443}{222} a^{2} + \frac{1193}{222} a + \frac{61}{37} \) (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:  \( 8765.05274681 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2$ (as 16T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), 4.0.1008.2, 4.4.49392.1, 4.0.1008.1, 4.0.49392.1, 4.0.1372.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 4.0.12348.1, 8.0.152473104.1, 8.0.49787136.3, 8.0.2439569664.7

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$