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Properties

Label 16.0.10019302138...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 41^{2}\cdot 61^{2}$
Root discriminant $15.40$
Ramified primes $3, 5, 41, 61$
Class number $1$
Class group Trivial
Galois group 16T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1831, 5701, 5643, -1338, -6198, -2537, 2092, 1876, 135, -478, -304, 27, 98, 8, -15, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 15*x^14 + 8*x^13 + 98*x^12 + 27*x^11 - 304*x^10 - 478*x^9 + 135*x^8 + 1876*x^7 + 2092*x^6 - 2537*x^5 - 6198*x^4 - 1338*x^3 + 5643*x^2 + 5701*x + 1831)
 
gp: K = bnfinit(x^16 - x^15 - 15*x^14 + 8*x^13 + 98*x^12 + 27*x^11 - 304*x^10 - 478*x^9 + 135*x^8 + 1876*x^7 + 2092*x^6 - 2537*x^5 - 6198*x^4 - 1338*x^3 + 5643*x^2 + 5701*x + 1831, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 15 x^{14} + 8 x^{13} + 98 x^{12} + 27 x^{11} - 304 x^{10} - 478 x^{9} + 135 x^{8} + 1876 x^{7} + 2092 x^{6} - 2537 x^{5} - 6198 x^{4} - 1338 x^{3} + 5643 x^{2} + 5701 x + 1831 \)

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:  \(10019302138916015625=3^{8}\cdot 5^{12}\cdot 41^{2}\cdot 61^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.40$
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:  $3, 5, 41, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} + \frac{2}{5} a^{10} + \frac{3}{10} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{3}{10} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{10}$, $\frac{1}{10} a^{13} + \frac{2}{5} a^{11} + \frac{3}{10} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{3}{10} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{2} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{10} a$, $\frac{1}{10} a^{14} + \frac{3}{10} a^{11} + \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{10} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{797458034933290} a^{15} - \frac{34452477603367}{797458034933290} a^{14} + \frac{3410665003425}{159491606986658} a^{13} + \frac{15274389554117}{398729017466645} a^{12} - \frac{45070487960201}{797458034933290} a^{11} + \frac{273604647067551}{797458034933290} a^{10} + \frac{113601618729894}{398729017466645} a^{9} + \frac{193912057703471}{797458034933290} a^{8} - \frac{71474402250347}{159491606986658} a^{7} - \frac{86447660463906}{398729017466645} a^{6} - \frac{222095910176951}{797458034933290} a^{5} - \frac{75599659286783}{159491606986658} a^{4} - \frac{168871961007687}{398729017466645} a^{3} + \frac{340491944158481}{797458034933290} a^{2} - \frac{97814526587181}{797458034933290} a - \frac{282476310646041}{797458034933290}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{37918065648477}{79745803493329} a^{15} - \frac{98423729600317}{79745803493329} a^{14} - \frac{4118711628640943}{797458034933290} a^{13} + \frac{9614907975997109}{797458034933290} a^{12} + \frac{10909657485571839}{398729017466645} a^{11} - \frac{24658490557919913}{797458034933290} a^{10} - \frac{75894155370888001}{797458034933290} a^{9} - \frac{29955004619289034}{398729017466645} a^{8} + \frac{146904520790949727}{797458034933290} a^{7} + \frac{95362979537864009}{159491606986658} a^{6} + \frac{15538549247651327}{398729017466645} a^{5} - \frac{1012635238404142459}{797458034933290} a^{4} - \frac{731423142894640927}{797458034933290} a^{3} + \frac{66305369001424241}{79745803493329} a^{2} + \frac{216195171486460389}{159491606986658} a + \frac{433756932965217939}{797458034933290} \) (order $30$)
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:  \( 9841.14273053 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n797 are not computed
Character table for t16n797 is not computed

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{15})\)

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
3Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
61Data not computed