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Properties

Label 16.0.19195811213...9424.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 41^{6}\cdot 97^{2}$
Root discriminant $28.52$
Ramified primes $2, 41, 97$
Class number $24$ (GRH)
Class group $[24]$ (GRH)
Galois group 16T876

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3929, -932, 1800, -2840, 5011, -4468, 4490, -3192, 2239, -1252, 724, -316, 143, -52, 16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 16*x^14 - 52*x^13 + 143*x^12 - 316*x^11 + 724*x^10 - 1252*x^9 + 2239*x^8 - 3192*x^7 + 4490*x^6 - 4468*x^5 + 5011*x^4 - 2840*x^3 + 1800*x^2 - 932*x + 3929)
 
gp: K = bnfinit(x^16 - 4*x^15 + 16*x^14 - 52*x^13 + 143*x^12 - 316*x^11 + 724*x^10 - 1252*x^9 + 2239*x^8 - 3192*x^7 + 4490*x^6 - 4468*x^5 + 5011*x^4 - 2840*x^3 + 1800*x^2 - 932*x + 3929, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 16 x^{14} - 52 x^{13} + 143 x^{12} - 316 x^{11} + 724 x^{10} - 1252 x^{9} + 2239 x^{8} - 3192 x^{7} + 4490 x^{6} - 4468 x^{5} + 5011 x^{4} - 2840 x^{3} + 1800 x^{2} - 932 x + 3929 \)

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:  \(191958112137556655079424=2^{32}\cdot 41^{6}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.52$
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:  $2, 41, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} - \frac{2}{7} a^{12} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{3}{7} a^{8} + \frac{2}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{14} + \frac{3}{7} a^{12} - \frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{3046979802783252789594673} a^{15} - \frac{101741953892497233524668}{3046979802783252789594673} a^{14} + \frac{156963360115484933397369}{3046979802783252789594673} a^{13} - \frac{1296083619027279258418675}{3046979802783252789594673} a^{12} + \frac{1502476542947771148290622}{3046979802783252789594673} a^{11} - \frac{241706096849550636833224}{3046979802783252789594673} a^{10} + \frac{9126945462400931485080}{435282828969036112799239} a^{9} - \frac{452284697984597508045399}{3046979802783252789594673} a^{8} + \frac{251578087072796206024566}{3046979802783252789594673} a^{7} - \frac{941162178021143620759050}{3046979802783252789594673} a^{6} + \frac{1515706449581269019867657}{3046979802783252789594673} a^{5} - \frac{3503271531016163029910}{11764400782946921967547} a^{4} + \frac{407129854899790638288451}{3046979802783252789594673} a^{3} + \frac{41463844985953108208002}{179234106046073693505569} a^{2} - \frac{120203501979669346840568}{435282828969036112799239} a + \frac{92707349353656217633528}{3046979802783252789594673}$

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

Class group and class number

$C_{24}$, which has order $24$ (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:  \( -1 \) (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:  \( 2873.19936942 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T876:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 65 conjugacy class representatives for t16n876 are not computed
Character table for t16n876 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.0.10686103552.1, 8.8.282300416.1, 8.0.438130245632.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$