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Properties

Label 16.16.1045558275...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{40}\cdot 5^{12}\cdot 79^{4}$
Root discriminant $56.39$
Ramified primes $2, 5, 79$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4:C_2^2.C_2$ (as 16T317)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3321, -5400, -138960, -123492, 264382, 282880, -143096, -179028, 38794, 48352, -8176, -6084, 1154, 296, -64, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 64*x^14 + 296*x^13 + 1154*x^12 - 6084*x^11 - 8176*x^10 + 48352*x^9 + 38794*x^8 - 179028*x^7 - 143096*x^6 + 282880*x^5 + 264382*x^4 - 123492*x^3 - 138960*x^2 - 5400*x + 3321)
 
gp: K = bnfinit(x^16 - 4*x^15 - 64*x^14 + 296*x^13 + 1154*x^12 - 6084*x^11 - 8176*x^10 + 48352*x^9 + 38794*x^8 - 179028*x^7 - 143096*x^6 + 282880*x^5 + 264382*x^4 - 123492*x^3 - 138960*x^2 - 5400*x + 3321, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 64 x^{14} + 296 x^{13} + 1154 x^{12} - 6084 x^{11} - 8176 x^{10} + 48352 x^{9} + 38794 x^{8} - 179028 x^{7} - 143096 x^{6} + 282880 x^{5} + 264382 x^{4} - 123492 x^{3} - 138960 x^{2} - 5400 x + 3321 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10455582754471936000000000000=2^{40}\cdot 5^{12}\cdot 79^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.39$
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:  $2, 5, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{5} + \frac{3}{10} a^{4} + \frac{3}{10} a^{3} - \frac{1}{2} a^{2} - \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{150} a^{13} + \frac{1}{75} a^{12} + \frac{11}{150} a^{11} - \frac{17}{75} a^{10} + \frac{29}{150} a^{9} - \frac{4}{25} a^{8} + \frac{1}{3} a^{7} + \frac{11}{75} a^{6} - \frac{11}{150} a^{5} + \frac{7}{25} a^{4} - \frac{71}{150} a^{3} + \frac{26}{75} a^{2} + \frac{37}{150} a - \frac{7}{25}$, $\frac{1}{4950} a^{14} - \frac{1}{495} a^{13} - \frac{29}{2475} a^{12} - \frac{398}{2475} a^{11} - \frac{343}{4950} a^{10} + \frac{58}{825} a^{9} - \frac{326}{2475} a^{8} - \frac{589}{2475} a^{7} + \frac{5}{18} a^{6} + \frac{14}{825} a^{5} + \frac{94}{495} a^{4} + \frac{722}{2475} a^{3} + \frac{1513}{4950} a^{2} + \frac{18}{275} a + \frac{63}{275}$, $\frac{1}{4604641167935573438850} a^{15} + \frac{96643809614111803}{2302320583967786719425} a^{14} - \frac{5354218352253489161}{2302320583967786719425} a^{13} - \frac{68646558852414408506}{2302320583967786719425} a^{12} - \frac{568085849749912436234}{2302320583967786719425} a^{11} - \frac{743047084305268481}{85271132739547656275} a^{10} + \frac{5109795643829036063}{460464116793557343885} a^{9} + \frac{508860248786492187413}{2302320583967786719425} a^{8} + \frac{1249522141300829736277}{4604641167935573438850} a^{7} + \frac{37480197108758909399}{85271132739547656275} a^{6} + \frac{297529687923203410634}{2302320583967786719425} a^{5} + \frac{584362411829161628063}{2302320583967786719425} a^{4} + \frac{327883197919957127243}{2302320583967786719425} a^{3} - \frac{43923946362574324721}{767440194655928906475} a^{2} - \frac{7462519465651882022}{17054226547909531255} a - \frac{37344718677859498342}{85271132739547656275}$

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:  $15$
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:  \( 772655833.855 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2.C_2$ (as 16T317):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4:C_2^2.C_2$
Character table for $C_2^4:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.8000.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.8.6.1$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}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.2.1.1$x^{2} - 79$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} - 79$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} - 79$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} - 79$$2$$1$$1$$C_2$$[\ ]_{2}$