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Properties

Label 16.4.16182878367...3184.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{34}\cdot 3^{8}\cdot 13^{4}\cdot 709^{2}$
Root discriminant $32.59$
Ramified primes $2, 3, 13, 709$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1605

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15214, -21776, -10884, 19716, 11552, -14272, -5844, 6484, 1914, -1556, -596, 256, 94, -12, -14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 14*x^14 - 12*x^13 + 94*x^12 + 256*x^11 - 596*x^10 - 1556*x^9 + 1914*x^8 + 6484*x^7 - 5844*x^6 - 14272*x^5 + 11552*x^4 + 19716*x^3 - 10884*x^2 - 21776*x + 15214)
 
gp: K = bnfinit(x^16 - 14*x^14 - 12*x^13 + 94*x^12 + 256*x^11 - 596*x^10 - 1556*x^9 + 1914*x^8 + 6484*x^7 - 5844*x^6 - 14272*x^5 + 11552*x^4 + 19716*x^3 - 10884*x^2 - 21776*x + 15214, 1)
 

Normalized defining polynomial

\( x^{16} - 14 x^{14} - 12 x^{13} + 94 x^{12} + 256 x^{11} - 596 x^{10} - 1556 x^{9} + 1914 x^{8} + 6484 x^{7} - 5844 x^{6} - 14272 x^{5} + 11552 x^{4} + 19716 x^{3} - 10884 x^{2} - 21776 x + 15214 \)

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:  \(1618287836733993526493184=2^{34}\cdot 3^{8}\cdot 13^{4}\cdot 709^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.59$
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:  $2, 3, 13, 709$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{80776062946753290682565073450529} a^{15} - \frac{26077616146338744364174768576486}{80776062946753290682565073450529} a^{14} - \frac{9558815171430396242795762594249}{80776062946753290682565073450529} a^{13} + \frac{11889821574421900017641595681038}{80776062946753290682565073450529} a^{12} - \frac{39346008805600571688277225895780}{80776062946753290682565073450529} a^{11} + \frac{12101081216789992104447805879767}{80776062946753290682565073450529} a^{10} + \frac{19511703580705260823003298199799}{80776062946753290682565073450529} a^{9} - \frac{28756055799471258841646936590724}{80776062946753290682565073450529} a^{8} + \frac{34720982677337330984894590467609}{80776062946753290682565073450529} a^{7} + \frac{33104460213420606070075281486560}{80776062946753290682565073450529} a^{6} + \frac{23345307442256020285236418679675}{80776062946753290682565073450529} a^{5} + \frac{12659888666624249074415627640222}{80776062946753290682565073450529} a^{4} + \frac{36475935515900270904424216874394}{80776062946753290682565073450529} a^{3} + \frac{17077822636555901213462144521221}{80776062946753290682565073450529} a^{2} - \frac{17964311786379859736951949706853}{80776062946753290682565073450529} a - \frac{35652560066359411487898434820845}{80776062946753290682565073450529}$

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:  \( 417647.378858 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1605:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 124 conjugacy class representatives for t16n1605 are not computed
Character table for t16n1605 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.7488.1, 8.4.224280576.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
3Data not computed
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
709Data not computed