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Properties

Label 16.0.38982341165...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{12}\cdot 29^{6}$
Root discriminant $39.76$
Ramified primes $2, 5, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T456)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37845, -238380, 684110, -1152650, 1227621, -824978, 319616, -45882, -10048, 400, 3730, -1526, 113, 54, 0, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 54*x^13 + 113*x^12 - 1526*x^11 + 3730*x^10 + 400*x^9 - 10048*x^8 - 45882*x^7 + 319616*x^6 - 824978*x^5 + 1227621*x^4 - 1152650*x^3 + 684110*x^2 - 238380*x + 37845)
 
gp: K = bnfinit(x^16 - 6*x^15 + 54*x^13 + 113*x^12 - 1526*x^11 + 3730*x^10 + 400*x^9 - 10048*x^8 - 45882*x^7 + 319616*x^6 - 824978*x^5 + 1227621*x^4 - 1152650*x^3 + 684110*x^2 - 238380*x + 37845, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 54 x^{13} + 113 x^{12} - 1526 x^{11} + 3730 x^{10} + 400 x^{9} - 10048 x^{8} - 45882 x^{7} + 319616 x^{6} - 824978 x^{5} + 1227621 x^{4} - 1152650 x^{3} + 684110 x^{2} - 238380 x + 37845 \)

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:  \(38982341165056000000000000=2^{28}\cdot 5^{12}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.76$
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:  $2, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{36} a^{12} + \frac{1}{18} a^{11} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} + \frac{4}{9} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{9} a^{4} - \frac{5}{18} a^{3} + \frac{1}{18} a^{2} + \frac{2}{9} a - \frac{1}{12}$, $\frac{1}{1044} a^{13} + \frac{1}{116} a^{12} + \frac{43}{522} a^{11} - \frac{3}{58} a^{10} - \frac{79}{174} a^{9} + \frac{112}{261} a^{8} - \frac{127}{522} a^{7} + \frac{6}{29} a^{6} - \frac{139}{522} a^{5} - \frac{43}{174} a^{4} + \frac{127}{261} a^{3} + \frac{17}{522} a^{2} + \frac{1}{36} a + \frac{1}{12}$, $\frac{1}{2088} a^{14} + \frac{5}{2088} a^{12} + \frac{3}{29} a^{11} + \frac{1}{174} a^{10} - \frac{253}{1044} a^{9} - \frac{55}{1044} a^{8} + \frac{23}{116} a^{7} + \frac{455}{1044} a^{6} - \frac{37}{87} a^{5} - \frac{151}{1044} a^{4} - \frac{181}{1044} a^{3} - \frac{277}{2088} a^{2} - \frac{1}{12} a + \frac{1}{8}$, $\frac{1}{6552525771019698567192} a^{15} + \frac{74943112198693361}{6552525771019698567192} a^{14} - \frac{1136122751260825951}{6552525771019698567192} a^{13} - \frac{83951112902132306167}{6552525771019698567192} a^{12} - \frac{80643913547151873061}{1638131442754924641798} a^{11} + \frac{163531337628940954889}{3276262885509849283596} a^{10} - \frac{23626657700786981456}{273021907125820773633} a^{9} + \frac{253632822072361470217}{819065721377462320899} a^{8} + \frac{275058311576956704395}{1638131442754924641798} a^{7} + \frac{27542409383136356855}{112974582258960320124} a^{6} - \frac{619051435137929838751}{3276262885509849283596} a^{5} - \frac{202124125263143834512}{819065721377462320899} a^{4} - \frac{705156781298669748173}{2184175257006566189064} a^{3} - \frac{31700107865990695357}{75316388172640213416} a^{2} - \frac{40850092194523436705}{225949164517920640248} a + \frac{279544453274851307}{2597116833539317704}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( \frac{55123363128404}{247846562116731} a^{15} - \frac{267951486117880}{247846562116731} a^{14} - \frac{305376940217780}{247846562116731} a^{13} + \frac{5258853111483745}{495693124233462} a^{12} + \frac{9225662265965405}{247846562116731} a^{11} - \frac{73615734582838435}{247846562116731} a^{10} + \frac{121722632324843230}{247846562116731} a^{9} + \frac{160878822780392435}{247846562116731} a^{8} - \frac{123591596324526275}{82615520705577} a^{7} - \frac{101799769798107020}{8546433176439} a^{6} + \frac{14255920841732407751}{247846562116731} a^{5} - \frac{29227886040785241865}{247846562116731} a^{4} + \frac{34346617815261985430}{247846562116731} a^{3} - \frac{93359985918934470}{949603686271} a^{2} + \frac{341916017178163705}{8546433176439} a - \frac{1449644994385133}{196469728194} \) (order $4$)
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:  \( 2204103.04484 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T456):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), 4.4.58000.1, 4.0.3625.1, \(\Q(i, \sqrt{5})\), 8.0.3364000000.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.16.30$x^{8} + 8 x^{7} + 20$$4$$2$$16$$C_2^3: C_4$$[2, 2, 3]^{4}$
2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{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}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$