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Properties

Label 16.16.1623938554...3136.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{26}\cdot 193^{8}\cdot 257^{10}$
Root discriminant $1374.55$
Ramified primes $2, 193, 257$
Class number $24$ (GRH)
Class group $[2, 12]$ (GRH)
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1009289184953344, 0, -511323407119984, 0, 40283145849176, 0, -1264961735700, 0, 18976436546, 0, -147646866, 0, 606551, 0, -1244, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 1244*x^14 + 606551*x^12 - 147646866*x^10 + 18976436546*x^8 - 1264961735700*x^6 + 40283145849176*x^4 - 511323407119984*x^2 + 1009289184953344)
 
gp: K = bnfinit(x^16 - 1244*x^14 + 606551*x^12 - 147646866*x^10 + 18976436546*x^8 - 1264961735700*x^6 + 40283145849176*x^4 - 511323407119984*x^2 + 1009289184953344, 1)
 

Normalized defining polynomial

\( x^{16} - 1244 x^{14} + 606551 x^{12} - 147646866 x^{10} + 18976436546 x^{8} - 1264961735700 x^{6} + 40283145849176 x^{4} - 511323407119984 x^{2} + 1009289184953344 \)

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:  \(162393855437479514050983820757859401106269862363136=2^{26}\cdot 193^{8}\cdot 257^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1374.55$
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:  $2, 193, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{254944} a^{12} - \frac{2529}{254944} a^{10} - \frac{5767}{127472} a^{8} - \frac{13033}{127472} a^{6} - \frac{8979}{31868} a^{4} - \frac{47}{124} a^{2} - \frac{9}{31}$, $\frac{1}{254944} a^{13} - \frac{2529}{254944} a^{11} - \frac{5767}{127472} a^{9} - \frac{13033}{127472} a^{7} + \frac{6955}{31868} a^{5} - \frac{1}{2} a^{4} - \frac{47}{124} a^{3} - \frac{9}{31} a$, $\frac{1}{20009723338656940237739215232510997496384} a^{14} - \frac{1}{509888} a^{13} + \frac{3317322005670620443148425708809111}{20009723338656940237739215232510997496384} a^{12} - \frac{13405}{509888} a^{11} + \frac{198015391109891267935013049582855082013}{10004861669328470118869607616255498748192} a^{10} + \frac{6867}{127472} a^{9} + \frac{488371484543712957925655749178063089631}{10004861669328470118869607616255498748192} a^{8} + \frac{60835}{254944} a^{7} + \frac{410346142936487668931532996576566940951}{2501215417332117529717401904063874687048} a^{6} - \frac{5943}{127472} a^{5} - \frac{267991545745935652232793165603402541651}{2501215417332117529717401904063874687048} a^{4} + \frac{109}{248} a^{3} + \frac{459922928112111112049986105233943021}{2433088927365873083382686677104936466} a^{2} + \frac{49}{124} a - \frac{153634714713448286577724498176368342}{1216544463682936541691343338552468233}$, $\frac{1}{154595122514463520276773176886379966657062784} a^{15} - \frac{46512374720472793184878396507099022663}{38648780628615880069193294221594991664265696} a^{13} + \frac{2472521240681901952549551481260712719974815}{154595122514463520276773176886379966657062784} a^{11} - \frac{2736208769991564703836021631781288112687133}{77297561257231760138386588443189983328531392} a^{9} - \frac{18363009506853094367442573023976255003696231}{77297561257231760138386588443189983328531392} a^{7} - \frac{4047051896719487996957923747895052887671497}{38648780628615880069193294221594991664265696} a^{5} - \frac{642968117409304146407621423184083761349}{75192180211314941768858549069250956545264} a^{3} + \frac{15892126117846048068712351485995682672829}{37596090105657470884429274534625478272632} a$

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

Class group and class number

$C_{2}\times C_{12}$, 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:  $15$
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:  \( 803436545961000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{257}) \), \(\Q(\sqrt{193}) \), \(\Q(\sqrt{49601}) \), 4.4.19682073608.1, 4.4.528392.1, \(\Q(\sqrt{193}, \sqrt{257})\), 8.8.387384021510730137664.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.11.15$x^{4} + 30$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.4.10.2$x^{4} + 2 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
193Data not computed
257Data not computed