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Properties

Label 16.0.28389329970...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{10}\cdot 17^{6}\cdot 19^{6}$
Root discriminant $33.75$
Ramified primes $2, 5, 17, 19$
Class number $2$
Class group $[2]$
Galois group 16T1191

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![71, -19, 2282, 345, 20962, 8198, 15655, 5336, 4700, 691, 678, -66, 101, -20, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 - 20*x^13 + 101*x^12 - 66*x^11 + 678*x^10 + 691*x^9 + 4700*x^8 + 5336*x^7 + 15655*x^6 + 8198*x^5 + 20962*x^4 + 345*x^3 + 2282*x^2 - 19*x + 71)
 
gp: K = bnfinit(x^16 + 16*x^14 - 20*x^13 + 101*x^12 - 66*x^11 + 678*x^10 + 691*x^9 + 4700*x^8 + 5336*x^7 + 15655*x^6 + 8198*x^5 + 20962*x^4 + 345*x^3 + 2282*x^2 - 19*x + 71, 1)
 

Normalized defining polynomial

\( x^{16} + 16 x^{14} - 20 x^{13} + 101 x^{12} - 66 x^{11} + 678 x^{10} + 691 x^{9} + 4700 x^{8} + 5336 x^{7} + 15655 x^{6} + 8198 x^{5} + 20962 x^{4} + 345 x^{3} + 2282 x^{2} - 19 x + 71 \)

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:  \(2838932997008222500000000=2^{8}\cdot 5^{10}\cdot 17^{6}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.75$
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:  $2, 5, 17, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{9} - \frac{2}{15} a^{8} + \frac{2}{5} a^{7} + \frac{2}{15} a^{6} + \frac{2}{5} a^{5} + \frac{2}{15} a^{4} + \frac{1}{5} a^{2} - \frac{1}{15}$, $\frac{1}{15} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{3} a^{9} - \frac{1}{5} a^{8} - \frac{1}{15} a^{7} - \frac{1}{15} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{15} a + \frac{1}{5}$, $\frac{1}{2055} a^{14} - \frac{7}{685} a^{13} - \frac{41}{2055} a^{12} + \frac{113}{685} a^{11} - \frac{26}{2055} a^{10} + \frac{66}{137} a^{9} - \frac{333}{685} a^{8} + \frac{68}{685} a^{7} + \frac{149}{411} a^{6} - \frac{219}{685} a^{5} - \frac{113}{411} a^{4} + \frac{76}{685} a^{3} - \frac{568}{2055} a^{2} - \frac{192}{685} a + \frac{89}{2055}$, $\frac{1}{334431986268033874044645555} a^{15} - \frac{51916522424013731970328}{334431986268033874044645555} a^{14} - \frac{9273167750069434532815252}{334431986268033874044645555} a^{13} - \frac{5076705413116339760207573}{334431986268033874044645555} a^{12} + \frac{971388727783202117896816}{334431986268033874044645555} a^{11} - \frac{34224540546446719030184986}{334431986268033874044645555} a^{10} + \frac{149592280336183179498275212}{334431986268033874044645555} a^{9} - \frac{28627747689657580236189946}{334431986268033874044645555} a^{8} - \frac{39402773258594131687544833}{111477328756011291348215185} a^{7} + \frac{2985056707259555667370309}{22295465751202258269643037} a^{6} - \frac{154606439938395559934204792}{334431986268033874044645555} a^{5} - \frac{135420521238859180784257717}{334431986268033874044645555} a^{4} + \frac{163210644028859048254295732}{334431986268033874044645555} a^{3} + \frac{11342351327742602434072814}{66886397253606774808929111} a^{2} - \frac{151014817889612017807277666}{334431986268033874044645555} a - \frac{152698158631108975895406223}{334431986268033874044645555}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 182966.399368 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1191:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1191 are not computed
Character table for t16n1191 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.475.1, 8.0.65205625.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
17.8.6.4$x^{8} + 136 x^{4} + 7803$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$