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Properties

Label 16.4.42287613544...3041.1
Degree $16$
Signature $[4, 6]$
Discriminant $37^{4}\cdot 41^{12}$
Root discriminant $39.96$
Ramified primes $37, 41$
Class number $1$ (GRH)
Class group Trivial (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![5888, -22400, 21952, 5376, -14480, 15648, -7256, 908, 677, -412, -32, -52, 130, -84, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 130*x^12 - 52*x^11 - 32*x^10 - 412*x^9 + 677*x^8 + 908*x^7 - 7256*x^6 + 15648*x^5 - 14480*x^4 + 5376*x^3 + 21952*x^2 - 22400*x + 5888)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 130*x^12 - 52*x^11 - 32*x^10 - 412*x^9 + 677*x^8 + 908*x^7 - 7256*x^6 + 15648*x^5 - 14480*x^4 + 5376*x^3 + 21952*x^2 - 22400*x + 5888, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 130 x^{12} - 52 x^{11} - 32 x^{10} - 412 x^{9} + 677 x^{8} + 908 x^{7} - 7256 x^{6} + 15648 x^{5} - 14480 x^{4} + 5376 x^{3} + 21952 x^{2} - 22400 x + 5888 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(42287613544824591671753041=37^{4}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.96$
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:  $37, 41$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{5} - \frac{1}{16} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{9} + \frac{1}{16} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{9} - \frac{1}{16} a^{7} + \frac{1}{32} a^{6} + \frac{3}{32} a^{5} - \frac{1}{16} a^{4}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} - \frac{1}{32} a^{7} + \frac{1}{32} a^{5}$, $\frac{1}{7872} a^{12} - \frac{1}{1312} a^{11} + \frac{121}{7872} a^{10} - \frac{29}{3936} a^{9} - \frac{15}{2624} a^{8} + \frac{77}{1312} a^{7} + \frac{1}{2624} a^{6} + \frac{11}{1312} a^{5} + \frac{25}{246} a^{4} - \frac{9}{164} a^{3} - \frac{79}{492} a^{2} + \frac{11}{246} a - \frac{4}{123}$, $\frac{1}{7872} a^{13} + \frac{85}{7872} a^{11} - \frac{35}{3936} a^{10} - \frac{49}{2624} a^{9} + \frac{1}{41} a^{8} + \frac{105}{2624} a^{7} + \frac{55}{1312} a^{6} - \frac{263}{3936} a^{5} + \frac{77}{656} a^{4} + \frac{5}{492} a^{3} + \frac{10}{123} a^{2} - \frac{65}{246} a - \frac{8}{41}$, $\frac{1}{6439296} a^{14} - \frac{7}{6439296} a^{13} - \frac{229}{6439296} a^{12} + \frac{1465}{6439296} a^{11} - \frac{76027}{6439296} a^{10} - \frac{35917}{6439296} a^{9} + \frac{9123}{2146432} a^{8} + \frac{40809}{2146432} a^{7} + \frac{43375}{3219648} a^{6} - \frac{24235}{201228} a^{5} - \frac{35581}{804912} a^{4} - \frac{9349}{100614} a^{3} + \frac{217}{16769} a^{2} + \frac{11323}{50307} a - \frac{20321}{50307}$, $\frac{1}{29833258368} a^{15} + \frac{2309}{29833258368} a^{14} - \frac{4069}{264011136} a^{13} - \frac{1436879}{29833258368} a^{12} + \frac{62839501}{29833258368} a^{11} + \frac{21333757}{9944419456} a^{10} - \frac{127622777}{9944419456} a^{9} - \frac{96748295}{9944419456} a^{8} - \frac{315442433}{14916629184} a^{7} - \frac{54558673}{3729157296} a^{6} - \frac{121794583}{7458314592} a^{5} + \frac{98845169}{1864578648} a^{4} + \frac{211528033}{1864578648} a^{3} + \frac{130015835}{932289324} a^{2} - \frac{19427658}{77690777} a + \frac{26249003}{77690777}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:  \( 4824829.07092 \) (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

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.4.175753856917.1 x2, 8.4.6502892705929.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ R R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$