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Properties

Label 16.0.51399544780...7056.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 7^{8}$
Root discriminant $17.06$
Ramified primes $2, 3, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2\wr C_2$ (as 16T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![93, -366, 1128, -2394, 3955, -5216, 5834, -5664, 4876, -3634, 2302, -1192, 502, -168, 44, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 502*x^12 - 1192*x^11 + 2302*x^10 - 3634*x^9 + 4876*x^8 - 5664*x^7 + 5834*x^6 - 5216*x^5 + 3955*x^4 - 2394*x^3 + 1128*x^2 - 366*x + 93)
 
gp: K = bnfinit(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 502*x^12 - 1192*x^11 + 2302*x^10 - 3634*x^9 + 4876*x^8 - 5664*x^7 + 5834*x^6 - 5216*x^5 + 3955*x^4 - 2394*x^3 + 1128*x^2 - 366*x + 93, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 44 x^{14} - 168 x^{13} + 502 x^{12} - 1192 x^{11} + 2302 x^{10} - 3634 x^{9} + 4876 x^{8} - 5664 x^{7} + 5834 x^{6} - 5216 x^{5} + 3955 x^{4} - 2394 x^{3} + 1128 x^{2} - 366 x + 93 \)

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:  \(51399544780206637056=2^{24}\cdot 3^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.06$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{1407} a^{12} - \frac{2}{469} a^{11} + \frac{305}{1407} a^{10} - \frac{3}{67} a^{9} - \frac{102}{469} a^{8} + \frac{43}{469} a^{7} - \frac{178}{1407} a^{6} + \frac{102}{469} a^{5} - \frac{17}{201} a^{4} + \frac{225}{469} a^{3} - \frac{202}{469} a^{2} - \frac{46}{469} a - \frac{177}{469}$, $\frac{1}{1407} a^{13} + \frac{269}{1407} a^{11} + \frac{120}{469} a^{10} - \frac{228}{469} a^{9} - \frac{100}{469} a^{8} + \frac{596}{1407} a^{7} + \frac{215}{469} a^{6} + \frac{310}{1407} a^{5} - \frac{13}{469} a^{4} + \frac{30}{67} a^{3} + \frac{149}{469} a^{2} + \frac{16}{469} a - \frac{124}{469}$, $\frac{1}{66129} a^{14} - \frac{1}{9447} a^{13} + \frac{5}{66129} a^{12} + \frac{61}{66129} a^{11} - \frac{5396}{22043} a^{10} + \frac{4695}{22043} a^{9} - \frac{29080}{66129} a^{8} + \frac{4681}{9447} a^{7} + \frac{27310}{66129} a^{6} + \frac{7055}{66129} a^{5} - \frac{1141}{3149} a^{4} + \frac{8222}{22043} a^{3} - \frac{2572}{22043} a^{2} - \frac{9666}{22043} a + \frac{7731}{22043}$, $\frac{1}{103756401} a^{15} + \frac{37}{4940781} a^{14} + \frac{15338}{103756401} a^{13} - \frac{2963}{14822343} a^{12} + \frac{656306}{4940781} a^{11} - \frac{4786477}{103756401} a^{10} + \frac{40255268}{103756401} a^{9} - \frac{1333052}{4940781} a^{8} + \frac{35465848}{103756401} a^{7} + \frac{43149767}{103756401} a^{6} + \frac{14294605}{34585467} a^{5} - \frac{335285}{14822343} a^{4} - \frac{7729891}{34585467} a^{3} - \frac{13724555}{34585467} a^{2} - \frac{1592495}{11528489} a + \frac{10786813}{34585467}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{260}{9447} a^{14} + \frac{1820}{9447} a^{13} - \frac{3128}{3149} a^{12} + \frac{32644}{9447} a^{11} - \frac{89458}{9447} a^{10} + \frac{63810}{3149} a^{9} - \frac{328855}{9447} a^{8} + \frac{451564}{9447} a^{7} - \frac{180458}{3149} a^{6} + \frac{566648}{9447} a^{5} - \frac{526343}{9447} a^{4} + \frac{130782}{3149} a^{3} - \frac{78956}{3149} a^{2} + \frac{32030}{3149} a - \frac{9223}{3149} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8148.71784457 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2$ (as 16T39):

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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-21}) \), 4.0.3024.2, 4.0.189.1, 4.0.1008.1 x2, 4.2.9408.2 x2, \(\Q(\sqrt{-3}, \sqrt{7})\), 4.0.432.1, 4.0.21168.1, 8.0.7169347584.5, 8.0.448084224.7, 8.0.796594176.14, 8.0.146313216.1, 8.0.9144576.2, 8.0.7169347584.8, 8.0.448084224.1

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 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 ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$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}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$