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Properties

Label 16.0.49642480033...2361.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{15}\cdot 53^{3}$
Root discriminant $62.16$
Ramified primes $37, 53$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59321, 201961, 351229, 373741, 296303, 214177, 127675, 60204, 27113, 9348, 2685, 904, 218, 44, 25, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 25*x^14 + 44*x^13 + 218*x^12 + 904*x^11 + 2685*x^10 + 9348*x^9 + 27113*x^8 + 60204*x^7 + 127675*x^6 + 214177*x^5 + 296303*x^4 + 373741*x^3 + 351229*x^2 + 201961*x + 59321)
 
gp: K = bnfinit(x^16 - 2*x^15 + 25*x^14 + 44*x^13 + 218*x^12 + 904*x^11 + 2685*x^10 + 9348*x^9 + 27113*x^8 + 60204*x^7 + 127675*x^6 + 214177*x^5 + 296303*x^4 + 373741*x^3 + 351229*x^2 + 201961*x + 59321, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 25 x^{14} + 44 x^{13} + 218 x^{12} + 904 x^{11} + 2685 x^{10} + 9348 x^{9} + 27113 x^{8} + 60204 x^{7} + 127675 x^{6} + 214177 x^{5} + 296303 x^{4} + 373741 x^{3} + 351229 x^{2} + 201961 x + 59321 \)

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:  \(49642480033862407473413572361=37^{15}\cdot 53^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.16$
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:  $37, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{12} - \frac{1}{7} a^{11} + \frac{2}{7} a^{10} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7}$, $\frac{1}{16751029326646441929092464229079676639} a^{15} + \frac{622989922479999586282419478315538814}{16751029326646441929092464229079676639} a^{14} - \frac{3633715137749713227675969686213183310}{16751029326646441929092464229079676639} a^{13} + \frac{5751798744609486420005447090181834293}{16751029326646441929092464229079676639} a^{12} + \frac{5447621353255467601542097774578468435}{16751029326646441929092464229079676639} a^{11} - \frac{5386390193868116047780427199212077574}{16751029326646441929092464229079676639} a^{10} + \frac{916920413125000628479859332494287550}{2393004189520920275584637747011382377} a^{9} + \frac{2194817537974548335680442670944743596}{16751029326646441929092464229079676639} a^{8} - \frac{2364558990665791684425392611760998322}{16751029326646441929092464229079676639} a^{7} - \frac{7395437470478299811934113730508449392}{16751029326646441929092464229079676639} a^{6} + \frac{71732076799127248619635248415579795}{2393004189520920275584637747011382377} a^{5} - \frac{7089384809918558614305177773255915431}{16751029326646441929092464229079676639} a^{4} - \frac{4415989971124113440656217207672067160}{16751029326646441929092464229079676639} a^{3} - \frac{1539388564335945728392695871882712514}{16751029326646441929092464229079676639} a^{2} + \frac{3114296993675055059482639689535438961}{16751029326646441929092464229079676639} a - \frac{3187515344758739994248444576432633230}{16751029326646441929092464229079676639}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 12133117.6808 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 4.0.50653.1, 8.0.5031389488049.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R $16$

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
37Data not computed
53Data not computed