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Properties

Label 16.8.96652609448...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 5^{8}\cdot 29^{8}\cdot 139^{2}$
Root discriminant $31.56$
Ramified primes $2, 5, 29, 139$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3$ (as 16T364)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![400, 0, -1920, 0, 2664, 0, -1812, 0, 981, 0, -337, 0, 40, 0, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^14 + 40*x^12 - 337*x^10 + 981*x^8 - 1812*x^6 + 2664*x^4 - 1920*x^2 + 400)
 
gp: K = bnfinit(x^16 - x^14 + 40*x^12 - 337*x^10 + 981*x^8 - 1812*x^6 + 2664*x^4 - 1920*x^2 + 400, 1)
 

Normalized defining polynomial

\( x^{16} - x^{14} + 40 x^{12} - 337 x^{10} + 981 x^{8} - 1812 x^{6} + 2664 x^{4} - 1920 x^{2} + 400 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(966526094481948100000000=2^{8}\cdot 5^{8}\cdot 29^{8}\cdot 139^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.56$
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:  $2, 5, 29, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{10} + \frac{1}{5} a^{6} + \frac{7}{20} a^{4} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{11} - \frac{1}{40} a^{10} + \frac{1}{10} a^{7} - \frac{1}{10} a^{6} + \frac{7}{40} a^{5} - \frac{7}{40} a^{4} + \frac{1}{10} a^{3} - \frac{1}{10} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{80} a^{12} + \frac{1}{80} a^{10} - \frac{1}{8} a^{9} + \frac{1}{20} a^{8} - \frac{1}{4} a^{7} + \frac{11}{80} a^{6} + \frac{11}{80} a^{4} - \frac{3}{8} a^{3} + \frac{3}{10} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{80} a^{13} - \frac{1}{80} a^{11} + \frac{1}{20} a^{9} - \frac{1}{4} a^{8} + \frac{3}{80} a^{7} - \frac{3}{80} a^{5} - \frac{1}{2} a^{4} - \frac{3}{10} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{1638560} a^{14} - \frac{705}{327712} a^{12} - \frac{7729}{409640} a^{10} - \frac{1}{8} a^{9} + \frac{188227}{1638560} a^{8} - \frac{1}{4} a^{7} - \frac{31781}{148960} a^{6} + \frac{6283}{40964} a^{4} - \frac{3}{8} a^{3} + \frac{5507}{11704} a^{2} - \frac{1}{4} a - \frac{2537}{20482}$, $\frac{1}{16385600} a^{15} - \frac{64971}{16385600} a^{13} - \frac{17429}{1638560} a^{11} + \frac{761723}{16385600} a^{9} - \frac{1}{4} a^{8} + \frac{323861}{1489600} a^{7} + \frac{729879}{8192800} a^{5} + \frac{167983}{585200} a^{3} + \frac{1}{4} a^{2} - \frac{158689}{409640} a$

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:  $11$
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:  \( 1075418.12167 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T364):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), 4.4.4205.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 4.4.725.1 x2, 8.8.442050625.1, 8.4.983120590000.1, 8.4.983120590000.2

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

Sibling fields

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.2.1.2$x^{2} + 556$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.1.2$x^{2} + 556$$2$$1$$1$$C_2$$[\ ]_{2}$