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Properties

Label 16.8.60215786657...2784.3
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 3^{8}\cdot 17^{6}\cdot 97^{4}$
Root discriminant $62.91$
Ramified primes $2, 3, 17, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11840473, 6366400, -8480970, -2043696, 1813396, -500820, -58966, 110184, -427, 3132, -1490, -136, -36, -84, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 84*x^13 - 36*x^12 - 136*x^11 - 1490*x^10 + 3132*x^9 - 427*x^8 + 110184*x^7 - 58966*x^6 - 500820*x^5 + 1813396*x^4 - 2043696*x^3 - 8480970*x^2 + 6366400*x + 11840473)
 
gp: K = bnfinit(x^16 + 8*x^14 - 84*x^13 - 36*x^12 - 136*x^11 - 1490*x^10 + 3132*x^9 - 427*x^8 + 110184*x^7 - 58966*x^6 - 500820*x^5 + 1813396*x^4 - 2043696*x^3 - 8480970*x^2 + 6366400*x + 11840473, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} - 84 x^{13} - 36 x^{12} - 136 x^{11} - 1490 x^{10} + 3132 x^{9} - 427 x^{8} + 110184 x^{7} - 58966 x^{6} - 500820 x^{5} + 1813396 x^{4} - 2043696 x^{3} - 8480970 x^{2} + 6366400 x + 11840473 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(60215786657648562405359222784=2^{32}\cdot 3^{8}\cdot 17^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.91$
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:  $2, 3, 17, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{194} a^{14} + \frac{22}{97} a^{13} + \frac{5}{194} a^{12} - \frac{7}{97} a^{11} + \frac{16}{97} a^{10} - \frac{3}{194} a^{9} - \frac{19}{97} a^{8} + \frac{1}{97} a^{7} - \frac{43}{97} a^{6} - \frac{7}{194} a^{5} + \frac{2}{97} a^{4} - \frac{17}{97} a^{3} + \frac{41}{194} a^{2} + \frac{21}{194} a - \frac{71}{194}$, $\frac{1}{340442273709409610443854530521647215073572244842} a^{15} - \frac{753767191707490368283789470303071457474424597}{340442273709409610443854530521647215073572244842} a^{14} - \frac{22047730136611871902101037570443357413799961008}{170221136854704805221927265260823607536786122421} a^{13} - \frac{5830229870650157846119514687592549648813815049}{170221136854704805221927265260823607536786122421} a^{12} + \frac{36021740071007255917613180654127717010188488139}{170221136854704805221927265260823607536786122421} a^{11} - \frac{76779122741374039712276050968256639587665743887}{340442273709409610443854530521647215073572244842} a^{10} + \frac{7271614639320091665300464077258403953421803169}{340442273709409610443854530521647215073572244842} a^{9} + \frac{22189130332329287000671647823509275760352711110}{170221136854704805221927265260823607536786122421} a^{8} - \frac{30685878575911985258980898990973852415179510520}{170221136854704805221927265260823607536786122421} a^{7} - \frac{131431876011784538599331584493689545994810195313}{340442273709409610443854530521647215073572244842} a^{6} + \frac{138164213734125476321018144034024605584615382067}{340442273709409610443854530521647215073572244842} a^{5} + \frac{50340524690874901743095957995178314911451447910}{170221136854704805221927265260823607536786122421} a^{4} + \frac{157429095562900537677831681930356545428077409547}{340442273709409610443854530521647215073572244842} a^{3} + \frac{531121680370703207955260482535822326019480594}{24317305264957829317418180751546229648112303203} a^{2} - \frac{123637392289895007567635831283645337306429017061}{340442273709409610443854530521647215073572244842} a + \frac{47104423751894068510904163550830936959709881427}{170221136854704805221927265260823607536786122421}$

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:  $11$
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:  \( 388644996.742 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), 4.4.9792.1, 4.4.4352.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
3Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
97Data not computed