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Properties

Label 16.4.17925988595...0625.2
Degree $16$
Signature $[4, 6]$
Discriminant $3^{4}\cdot 5^{10}\cdot 29^{4}\cdot 179^{2}$
Root discriminant $15.97$
Ramified primes $3, 5, 29, 179$
Class number $1$
Class group Trivial
Galois group 16T1559

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 4, -5, 2, -24, 15, -36, 5, 36, 15, 24, 2, 5, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 4*x^14 + 5*x^13 + 2*x^12 + 24*x^11 + 15*x^10 + 36*x^9 + 5*x^8 - 36*x^7 + 15*x^6 - 24*x^5 + 2*x^4 - 5*x^3 + 4*x^2 + x + 1)
 
gp: K = bnfinit(x^16 - x^15 + 4*x^14 + 5*x^13 + 2*x^12 + 24*x^11 + 15*x^10 + 36*x^9 + 5*x^8 - 36*x^7 + 15*x^6 - 24*x^5 + 2*x^4 - 5*x^3 + 4*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 4 x^{14} + 5 x^{13} + 2 x^{12} + 24 x^{11} + 15 x^{10} + 36 x^{9} + 5 x^{8} - 36 x^{7} + 15 x^{6} - 24 x^{5} + 2 x^{4} - 5 x^{3} + 4 x^{2} + x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17925988595712890625=3^{4}\cdot 5^{10}\cdot 29^{4}\cdot 179^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.97$
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:  $3, 5, 29, 179$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{165} a^{12} - \frac{28}{165} a^{11} + \frac{1}{15} a^{10} + \frac{82}{165} a^{9} + \frac{4}{55} a^{8} - \frac{13}{165} a^{7} + \frac{16}{165} a^{6} + \frac{13}{165} a^{5} + \frac{4}{55} a^{4} - \frac{82}{165} a^{3} + \frac{1}{15} a^{2} + \frac{28}{165} a + \frac{1}{165}$, $\frac{1}{165} a^{13} + \frac{52}{165} a^{11} + \frac{4}{11} a^{10} - \frac{2}{165} a^{9} - \frac{7}{165} a^{8} - \frac{6}{55} a^{7} - \frac{34}{165} a^{6} + \frac{46}{165} a^{5} - \frac{76}{165} a^{4} + \frac{5}{33} a^{3} + \frac{2}{55} a^{2} - \frac{8}{33} a + \frac{28}{165}$, $\frac{1}{1155} a^{14} - \frac{1}{1155} a^{13} - \frac{2}{1155} a^{12} + \frac{40}{231} a^{11} - \frac{23}{165} a^{10} - \frac{4}{15} a^{9} + \frac{1}{1155} a^{8} - \frac{139}{1155} a^{7} + \frac{41}{1155} a^{6} - \frac{47}{165} a^{5} - \frac{31}{165} a^{4} + \frac{284}{1155} a^{3} - \frac{29}{231} a^{2} + \frac{41}{1155} a + \frac{83}{1155}$, $\frac{1}{1155} a^{15} - \frac{1}{385} a^{13} + \frac{2}{1155} a^{12} - \frac{248}{1155} a^{11} - \frac{3}{11} a^{10} - \frac{19}{105} a^{9} - \frac{12}{77} a^{8} + \frac{4}{33} a^{7} + \frac{41}{1155} a^{6} + \frac{53}{165} a^{5} + \frac{5}{231} a^{4} + \frac{41}{1155} a^{3} + \frac{10}{231} a^{2} + \frac{137}{385} a - \frac{113}{1155}$

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:  \( 1982.45131564 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1559:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1559 are not computed
Character table for t16n1559 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.2.94086875.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$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}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_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$$[\ ]_{2}^{2}$
$179$$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.4.2.1$x^{4} + 2327 x^{2} + 1570009$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$