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Properties

Label 16.8.32031814824...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 13^{10}\cdot 29^{6}$
Root discriminant $39.27$
Ramified primes $5, 13, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2531, -1020, -16078, 9802, 18694, -15236, -3368, 9006, -2531, -1608, 1282, -130, -181, 72, 1, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + x^14 + 72*x^13 - 181*x^12 - 130*x^11 + 1282*x^10 - 1608*x^9 - 2531*x^8 + 9006*x^7 - 3368*x^6 - 15236*x^5 + 18694*x^4 + 9802*x^3 - 16078*x^2 - 1020*x + 2531)
 
gp: K = bnfinit(x^16 - 6*x^15 + x^14 + 72*x^13 - 181*x^12 - 130*x^11 + 1282*x^10 - 1608*x^9 - 2531*x^8 + 9006*x^7 - 3368*x^6 - 15236*x^5 + 18694*x^4 + 9802*x^3 - 16078*x^2 - 1020*x + 2531, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + x^{14} + 72 x^{13} - 181 x^{12} - 130 x^{11} + 1282 x^{10} - 1608 x^{9} - 2531 x^{8} + 9006 x^{7} - 3368 x^{6} - 15236 x^{5} + 18694 x^{4} + 9802 x^{3} - 16078 x^{2} - 1020 x + 2531 \)

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:  \(32031814824091254112890625=5^{8}\cdot 13^{10}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.27$
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:  $5, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{20} a^{13} - \frac{1}{10} a^{12} - \frac{1}{5} a^{11} + \frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{3}{20} a^{8} + \frac{7}{20} a^{7} - \frac{1}{10} a^{6} + \frac{1}{20} a^{5} + \frac{2}{5} a^{4} - \frac{1}{10} a^{3} - \frac{3}{20} a^{2} - \frac{2}{5} a + \frac{7}{20}$, $\frac{1}{2300} a^{14} + \frac{1}{46} a^{13} + \frac{167}{2300} a^{12} + \frac{199}{2300} a^{11} + \frac{7}{460} a^{10} - \frac{59}{575} a^{9} + \frac{263}{1150} a^{8} + \frac{218}{575} a^{7} + \frac{571}{1150} a^{6} + \frac{157}{460} a^{5} - \frac{109}{575} a^{4} + \frac{33}{2300} a^{3} - \frac{146}{575} a^{2} - \frac{287}{1150} a + \frac{209}{2300}$, $\frac{1}{22987016977562922003500} a^{15} - \frac{380090679777698052}{5746754244390730500875} a^{14} - \frac{187963908273168680129}{11493508488781461001750} a^{13} + \frac{412575920253361033063}{22987016977562922003500} a^{12} + \frac{1403925220459218581043}{22987016977562922003500} a^{11} - \frac{1054743658315764332979}{5746754244390730500875} a^{10} + \frac{62121501851203763407}{11493508488781461001750} a^{9} + \frac{2202194496051471511989}{22987016977562922003500} a^{8} - \frac{1932060139663363280959}{22987016977562922003500} a^{7} - \frac{4917252034174356683951}{22987016977562922003500} a^{6} + \frac{7982551369120271680809}{22987016977562922003500} a^{5} + \frac{8432481202665097364221}{22987016977562922003500} a^{4} - \frac{4308378640325106438899}{11493508488781461001750} a^{3} - \frac{11325109687767649657077}{22987016977562922003500} a^{2} + \frac{7157532328295185483201}{22987016977562922003500} a - \frac{8949854653194642646397}{22987016977562922003500}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 3202856.65549 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n869
Character table for t16n869 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{65}) \), 4.4.725.1, 4.4.122525.1, \(\Q(\sqrt{5}, \sqrt{13})\), 8.8.15012375625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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$$[\ ]_{2}^{4}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$