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Properties

Label 16.16.5654307216...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{12}\cdot 5^{10}\cdot 29^{8}\cdot 41^{4}$
Root discriminant $62.66$
Ramified primes $2, 5, 29, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3$ (as 16T329)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![483025, -1991175, -4442610, 3252475, 4584026, -1743798, -1949561, 421830, 421718, -49347, -49046, 2658, 3017, -52, -90, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 90*x^14 - 52*x^13 + 3017*x^12 + 2658*x^11 - 49046*x^10 - 49347*x^9 + 421718*x^8 + 421830*x^7 - 1949561*x^6 - 1743798*x^5 + 4584026*x^4 + 3252475*x^3 - 4442610*x^2 - 1991175*x + 483025)
 
gp: K = bnfinit(x^16 - 90*x^14 - 52*x^13 + 3017*x^12 + 2658*x^11 - 49046*x^10 - 49347*x^9 + 421718*x^8 + 421830*x^7 - 1949561*x^6 - 1743798*x^5 + 4584026*x^4 + 3252475*x^3 - 4442610*x^2 - 1991175*x + 483025, 1)
 

Normalized defining polynomial

\( x^{16} - 90 x^{14} - 52 x^{13} + 3017 x^{12} + 2658 x^{11} - 49046 x^{10} - 49347 x^{9} + 421718 x^{8} + 421830 x^{7} - 1949561 x^{6} - 1743798 x^{5} + 4584026 x^{4} + 3252475 x^{3} - 4442610 x^{2} - 1991175 x + 483025 \)

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:  \(56543072165403532840000000000=2^{12}\cdot 5^{10}\cdot 29^{8}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.66$
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:  $2, 5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{25} a^{14} - \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{25} a^{11} + \frac{12}{25} a^{10} + \frac{3}{25} a^{9} + \frac{9}{25} a^{8} + \frac{3}{25} a^{7} + \frac{8}{25} a^{6} - \frac{2}{5} a^{5} + \frac{4}{25} a^{4} + \frac{12}{25} a^{3} + \frac{1}{25} a^{2} - \frac{1}{5}$, $\frac{1}{6646245917367555664758013031786167909975} a^{15} - \frac{567671523939963330479034398872767786}{47814718829982414854374194473281783525} a^{14} - \frac{662277367447456704363519434204320681874}{1329249183473511132951602606357233581995} a^{13} - \frac{674583755581282920168606930685963868137}{6646245917367555664758013031786167909975} a^{12} - \frac{88780721102229942053659131370714645589}{265849836694702226590320521271446716399} a^{11} + \frac{24191297339522707176700700443962591304}{265849836694702226590320521271446716399} a^{10} + \frac{159948757159457011695752608496337031202}{6646245917367555664758013031786167909975} a^{9} + \frac{14538297427015614244609219532059306632}{6646245917367555664758013031786167909975} a^{8} + \frac{2958317555827770109215460338726086616451}{6646245917367555664758013031786167909975} a^{7} - \frac{1347305312573677826911049340095461709262}{6646245917367555664758013031786167909975} a^{6} - \frac{755711889968886304192906463399190817531}{6646245917367555664758013031786167909975} a^{5} + \frac{1718908054667742216075556551816787789461}{6646245917367555664758013031786167909975} a^{4} + \frac{1263251105827293042403597905694699617073}{6646245917367555664758013031786167909975} a^{3} - \frac{2907288299680525818866217453482849741619}{6646245917367555664758013031786167909975} a^{2} + \frac{366892882073671338377097498194262019934}{1329249183473511132951602606357233581995} a + \frac{457110411465781705469652062658535716}{9562943765996482970874838894656356705}$

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:  \( 1432911513.54 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T329):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 4.4.725.1 x2, 4.4.4205.1 x2, 8.8.442050625.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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$$[\ ]^{4}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$