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Properties

Label 16.0.92257744828...8125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 13^{8}\cdot 29^{5}\cdot 109^{4}$
Root discriminant $74.61$
Ramified primes $5, 13, 29, 109$
Class number $104$ (GRH)
Class group $[2, 52]$ (GRH)
Galois group 16T1581

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![136081, -112901, 94930, -62084, -22955, 39596, -15648, 16906, 3804, -2396, 1082, -167, 33, -53, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 12*x^14 - 53*x^13 + 33*x^12 - 167*x^11 + 1082*x^10 - 2396*x^9 + 3804*x^8 + 16906*x^7 - 15648*x^6 + 39596*x^5 - 22955*x^4 - 62084*x^3 + 94930*x^2 - 112901*x + 136081)
 
gp: K = bnfinit(x^16 + 12*x^14 - 53*x^13 + 33*x^12 - 167*x^11 + 1082*x^10 - 2396*x^9 + 3804*x^8 + 16906*x^7 - 15648*x^6 + 39596*x^5 - 22955*x^4 - 62084*x^3 + 94930*x^2 - 112901*x + 136081, 1)
 

Normalized defining polynomial

\( x^{16} + 12 x^{14} - 53 x^{13} + 33 x^{12} - 167 x^{11} + 1082 x^{10} - 2396 x^{9} + 3804 x^{8} + 16906 x^{7} - 15648 x^{6} + 39596 x^{5} - 22955 x^{4} - 62084 x^{3} + 94930 x^{2} - 112901 x + 136081 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(922577448288361543921511328125=5^{8}\cdot 13^{8}\cdot 29^{5}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.61$
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:  $5, 13, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5857828503228986201524683449969611483105} a^{15} - \frac{234791111535418721353930777011697221546}{5857828503228986201524683449969611483105} a^{14} - \frac{2502823209092964726639718233196254130166}{5857828503228986201524683449969611483105} a^{13} + \frac{112287194783400404260400456234720133757}{532529863929907836502243949997237407555} a^{12} + \frac{17511116345772334882604162812839295275}{106505972785981567300448789999447481511} a^{11} + \frac{256619562748132014043813929825086880102}{1171565700645797240304936689993922296621} a^{10} - \frac{2628709110246902161884595272266030571548}{5857828503228986201524683449969611483105} a^{9} + \frac{2639825108016696551734132189754975197982}{5857828503228986201524683449969611483105} a^{8} + \frac{447190457475219361349642812685078856314}{5857828503228986201524683449969611483105} a^{7} - \frac{1256921924244739327405545231201988639746}{5857828503228986201524683449969611483105} a^{6} + \frac{218768160382522845528482606984575675312}{5857828503228986201524683449969611483105} a^{5} + \frac{1224280474965624776005298991653000231198}{5857828503228986201524683449969611483105} a^{4} - \frac{2887584319681105053999237139412061946566}{5857828503228986201524683449969611483105} a^{3} - \frac{68080241501057117979496919035406494418}{1171565700645797240304936689993922296621} a^{2} - \frac{1550813599027985992460410238313910318201}{5857828503228986201524683449969611483105} a + \frac{34520117082275592042046017172682779992}{106505972785981567300448789999447481511}$

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

Class group and class number

$C_{2}\times C_{52}$, which has order $104$ (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:  $7$
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:  \( 3788129.71185 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1581:

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 t16n1581 are not computed
Character table for t16n1581 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.2725.1, 4.0.13355225.1, 4.4.122525.1, 8.0.178362034800625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13Data not computed
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
$109$109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$