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Properties

Label 16.4.80528182519...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{16}\cdot 5^{8}\cdot 17^{6}\cdot 19^{4}$
Root discriminant $27.02$
Ramified primes $2, 5, 17, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1605

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 142, -1184, 2142, 103, -1748, -280, 1148, -148, -106, -3, -26, -13, 30, -4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 + 30*x^13 - 13*x^12 - 26*x^11 - 3*x^10 - 106*x^9 - 148*x^8 + 1148*x^7 - 280*x^6 - 1748*x^5 + 103*x^4 + 2142*x^3 - 1184*x^2 + 142*x - 19)
 
gp: K = bnfinit(x^16 - 4*x^15 - 4*x^14 + 30*x^13 - 13*x^12 - 26*x^11 - 3*x^10 - 106*x^9 - 148*x^8 + 1148*x^7 - 280*x^6 - 1748*x^5 + 103*x^4 + 2142*x^3 - 1184*x^2 + 142*x - 19, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 4 x^{14} + 30 x^{13} - 13 x^{12} - 26 x^{11} - 3 x^{10} - 106 x^{9} - 148 x^{8} + 1148 x^{7} - 280 x^{6} - 1748 x^{5} + 103 x^{4} + 2142 x^{3} - 1184 x^{2} + 142 x - 19 \)

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:  \(80528182519014400000000=2^{16}\cdot 5^{8}\cdot 17^{6}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.02$
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:  $2, 5, 17, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{25} a^{12} + \frac{2}{25} a^{11} + \frac{2}{25} a^{10} - \frac{4}{25} a^{9} + \frac{12}{25} a^{8} + \frac{2}{5} a^{7} + \frac{4}{25} a^{6} + \frac{9}{25} a^{5} - \frac{2}{5} a^{4} - \frac{12}{25} a^{3} - \frac{7}{25} a^{2} + \frac{8}{25} a + \frac{11}{25}$, $\frac{1}{2375} a^{13} + \frac{43}{2375} a^{12} + \frac{129}{2375} a^{11} + \frac{173}{2375} a^{10} - \frac{1082}{2375} a^{9} + \frac{177}{2375} a^{8} + \frac{54}{2375} a^{7} - \frac{892}{2375} a^{6} - \frac{1171}{2375} a^{5} - \frac{787}{2375} a^{4} - \frac{909}{2375} a^{3} - \frac{509}{2375} a^{2} + \frac{1104}{2375} a + \frac{9}{125}$, $\frac{1}{2375} a^{14} - \frac{2}{475} a^{12} - \frac{54}{2375} a^{11} + \frac{124}{2375} a^{10} - \frac{987}{2375} a^{9} - \frac{337}{2375} a^{8} + \frac{111}{2375} a^{7} + \frac{13}{95} a^{6} + \frac{356}{2375} a^{5} + \frac{1107}{2375} a^{4} + \frac{8}{2375} a^{3} + \frac{1046}{2375} a^{2} - \frac{371}{2375} a + \frac{3}{125}$, $\frac{1}{17728840625} a^{15} - \frac{582399}{3545768125} a^{14} + \frac{53053}{377209375} a^{13} + \frac{15545374}{17728840625} a^{12} + \frac{18811753}{17728840625} a^{11} - \frac{777611349}{17728840625} a^{10} - \frac{2399495194}{17728840625} a^{9} - \frac{1740716602}{17728840625} a^{8} - \frac{8505731366}{17728840625} a^{7} + \frac{8228364254}{17728840625} a^{6} - \frac{184104176}{933096875} a^{5} + \frac{5382262806}{17728840625} a^{4} + \frac{1816082407}{17728840625} a^{3} - \frac{1355753709}{3545768125} a^{2} - \frac{2380916339}{17728840625} a + \frac{98061689}{933096875}$

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:  \( 232607.556467 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1605:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 124 conjugacy class representatives for t16n1605 are not computed
Character table for t16n1605 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.475.1, 8.4.16692640000.6

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{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$$[\ ]_{2}^{2}$
$17$17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
17.8.6.4$x^{8} + 136 x^{4} + 7803$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$