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Properties

Label 16.4.53358065924...1216.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{74}\cdot 7^{10}$
Root discriminant $83.26$
Ramified primes $2, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![529, 0, 3448, 0, -8748, 0, -14184, 0, -5806, 0, -680, 0, 140, 0, 24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^14 + 140*x^12 - 680*x^10 - 5806*x^8 - 14184*x^6 - 8748*x^4 + 3448*x^2 + 529)
 
gp: K = bnfinit(x^16 + 24*x^14 + 140*x^12 - 680*x^10 - 5806*x^8 - 14184*x^6 - 8748*x^4 + 3448*x^2 + 529, 1)
 

Normalized defining polynomial

\( x^{16} + 24 x^{14} + 140 x^{12} - 680 x^{10} - 5806 x^{8} - 14184 x^{6} - 8748 x^{4} + 3448 x^{2} + 529 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5335806592471429065121743241216=2^{74}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.26$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{1}{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} - \frac{1}{8} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{6} - \frac{1}{8} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{7} - \frac{1}{8} a^{3}$, $\frac{1}{1544} a^{12} - \frac{11}{386} a^{10} - \frac{5}{386} a^{8} + \frac{51}{193} a^{6} + \frac{347}{1544} a^{4} - \frac{147}{386} a^{2} - \frac{311}{772}$, $\frac{1}{35512} a^{13} - \frac{102}{4439} a^{11} - \frac{2143}{35512} a^{9} + \frac{1209}{4439} a^{7} - \frac{39}{35512} a^{5} + \frac{988}{4439} a^{3} - \frac{15483}{35512} a$, $\frac{1}{13873046896} a^{14} - \frac{3219229}{13873046896} a^{12} - \frac{75716533}{13873046896} a^{10} + \frac{747864079}{13873046896} a^{8} - \frac{1}{2} a^{7} + \frac{2804663661}{13873046896} a^{6} - \frac{1}{2} a^{5} - \frac{5449218773}{13873046896} a^{4} - \frac{1}{2} a^{3} + \frac{4171527131}{13873046896} a^{2} - \frac{1}{2} a - \frac{208553419}{603175952}$, $\frac{1}{13873046896} a^{15} - \frac{93965}{13873046896} a^{13} + \frac{842329767}{13873046896} a^{11} - \frac{747184087}{13873046896} a^{9} - \frac{1650400171}{13873046896} a^{7} - \frac{1}{2} a^{6} - \frac{2102842345}{13873046896} a^{5} - \frac{1}{2} a^{4} - \frac{2340741729}{13873046896} a^{3} - \frac{1}{2} a^{2} - \frac{2895396151}{13873046896} a - \frac{1}{2}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 1223642059.99 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.100352.1, 8.4.36092757671936.11

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$7$7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$