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Properties

Label 16.4.28683595087...6176.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{44}\cdot 113^{4}$
Root discriminant $21.93$
Ramified primes $2, 113$
Class number $1$
Class group Trivial
Galois group 16T1174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, -224, 0, -240, 0, -688, 0, 72, 0, -40, 0, 4, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 4*x^12 - 40*x^10 + 72*x^8 - 688*x^6 - 240*x^4 - 224*x^2 + 16)
 
gp: K = bnfinit(x^16 - 4*x^14 + 4*x^12 - 40*x^10 + 72*x^8 - 688*x^6 - 240*x^4 - 224*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} + 4 x^{12} - 40 x^{10} + 72 x^{8} - 688 x^{6} - 240 x^{4} - 224 x^{2} + 16 \)

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:  \(2868359508763057586176=2^{44}\cdot 113^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.93$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{32} a^{10} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{32} a^{11} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} + \frac{1}{8} a^{3} + \frac{1}{4} a$, $\frac{1}{128} a^{12} + \frac{1}{64} a^{8} - \frac{1}{8} a^{7} - \frac{7}{32} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{7}{16}$, $\frac{1}{128} a^{13} + \frac{1}{64} a^{9} - \frac{7}{32} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{7}{16} a - \frac{1}{2}$, $\frac{1}{750208} a^{14} + \frac{2835}{750208} a^{12} - \frac{5153}{375104} a^{10} + \frac{11391}{375104} a^{8} - \frac{1}{8} a^{7} - \frac{3887}{187552} a^{6} - \frac{40129}{187552} a^{4} + \frac{1}{4} a^{3} - \frac{3017}{93776} a^{2} - \frac{1}{2} a - \frac{43397}{93776}$, $\frac{1}{1500416} a^{15} - \frac{1513}{750208} a^{13} + \frac{6569}{750208} a^{11} - \frac{8957}{375104} a^{9} - \frac{3887}{375104} a^{7} - \frac{1}{8} a^{6} + \frac{12171}{187552} a^{5} - \frac{1}{4} a^{4} - \frac{85071}{187552} a^{3} + \frac{1}{4} a^{2} - \frac{24629}{93776} a$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 155771.019729 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 70 conjugacy class representatives for t16n1174 are not computed
Character table for t16n1174 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7232.1, 8.8.26778533888.1, 8.2.836829184.2, 8.2.6694633472.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.4.11.16$x^{4} + 14$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.4.11.16$x^{4} + 14$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.8.22.99$x^{8} + 4 x^{7} + 4 x^{6} + 6 x^{4} + 12 x^{2} + 14$$8$$1$$22$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$113$113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$