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Properties

Label 16.16.3245499096...0912.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{67}\cdot 17^{6}\cdot 977^{4}$
Root discriminant $294.75$
Ramified primes $2, 17, 977$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T937

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2106522414576392, 0, -775693364293408, 0, 46480812563696, 0, -1193982821024, 0, 16124860236, 0, -122203072, 0, 517764, 0, -1136, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 1136*x^14 + 517764*x^12 - 122203072*x^10 + 16124860236*x^8 - 1193982821024*x^6 + 46480812563696*x^4 - 775693364293408*x^2 + 2106522414576392)
 
gp: K = bnfinit(x^16 - 1136*x^14 + 517764*x^12 - 122203072*x^10 + 16124860236*x^8 - 1193982821024*x^6 + 46480812563696*x^4 - 775693364293408*x^2 + 2106522414576392, 1)
 

Normalized defining polynomial

\( x^{16} - 1136 x^{14} + 517764 x^{12} - 122203072 x^{10} + 16124860236 x^{8} - 1193982821024 x^{6} + 46480812563696 x^{4} - 775693364293408 x^{2} + 2106522414576392 \)

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:  \(3245499096990365116593765338925510950912=2^{67}\cdot 17^{6}\cdot 977^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $294.75$
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:  $2, 17, 977$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{33218} a^{10} - \frac{1545}{16609} a^{8} + \frac{1931}{16609} a^{6} + \frac{1021}{16609} a^{4} - \frac{7385}{16609} a^{2}$, $\frac{1}{66436} a^{11} - \frac{1545}{33218} a^{9} + \frac{1931}{33218} a^{7} - \frac{7794}{16609} a^{5} + \frac{4612}{16609} a^{3}$, $\frac{1}{64907972} a^{12} + \frac{409}{32453986} a^{10} + \frac{2438092}{16226993} a^{8} + \frac{2835276}{16226993} a^{6} + \frac{3145667}{16226993} a^{4} + \frac{258}{16609} a^{2}$, $\frac{1}{64907972} a^{13} - \frac{159}{64907972} a^{11} + \frac{6385649}{32453986} a^{9} + \frac{3783965}{32453986} a^{7} - \frac{321564}{954529} a^{5} - \frac{4354}{16609} a^{3}$, $\frac{1}{382303280110321456371147216933198134884} a^{14} + \frac{28424297503095769923410069527}{382303280110321456371147216933198134884} a^{12} + \frac{590869314751921145958306774664255}{95575820027580364092786804233299533721} a^{10} - \frac{15479499220298147015894761470474156897}{191151640055160728185573608466599067442} a^{8} + \frac{22880745958461252516000155137572360947}{95575820027580364092786804233299533721} a^{6} + \frac{43704812557013569291658176787702216}{97825813743685121896404098498771273} a^{4} - \frac{386686562119710787692952079795}{5889927975416046835836239297897} a^{2} - \frac{18471775331405263747470302}{354622672973450950438692233}$, $\frac{1}{382303280110321456371147216933198134884} a^{15} + \frac{28424297503095769923410069527}{382303280110321456371147216933198134884} a^{13} + \frac{590869314751921145958306774664255}{95575820027580364092786804233299533721} a^{11} - \frac{15479499220298147015894761470474156897}{191151640055160728185573608466599067442} a^{9} + \frac{22880745958461252516000155137572360947}{95575820027580364092786804233299533721} a^{7} + \frac{43704812557013569291658176787702216}{97825813743685121896404098498771273} a^{5} - \frac{386686562119710787692952079795}{5889927975416046835836239297897} a^{3} - \frac{18471775331405263747470302}{354622672973450950438692233} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 122365846131000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T937:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n937
Character table for t16n937 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.4352.1, 8.8.9697230848.1

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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
977Data not computed