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Properties

Label 16.0.17646465600...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 41^{2}$
Root discriminant $18.43$
Ramified primes $2, 3, 5, 41$
Class number $2$
Class group $[2]$
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![76, 164, -176, -698, 663, 968, -1028, -432, 691, -8, -191, 22, 25, 6, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 5*x^14 + 6*x^13 + 25*x^12 + 22*x^11 - 191*x^10 - 8*x^9 + 691*x^8 - 432*x^7 - 1028*x^6 + 968*x^5 + 663*x^4 - 698*x^3 - 176*x^2 + 164*x + 76)
 
gp: K = bnfinit(x^16 - 2*x^15 - 5*x^14 + 6*x^13 + 25*x^12 + 22*x^11 - 191*x^10 - 8*x^9 + 691*x^8 - 432*x^7 - 1028*x^6 + 968*x^5 + 663*x^4 - 698*x^3 - 176*x^2 + 164*x + 76, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 5 x^{14} + 6 x^{13} + 25 x^{12} + 22 x^{11} - 191 x^{10} - 8 x^{9} + 691 x^{8} - 432 x^{7} - 1028 x^{6} + 968 x^{5} + 663 x^{4} - 698 x^{3} - 176 x^{2} + 164 x + 76 \)

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:  \(176464656000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 41^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.43$
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:  $2, 3, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{84} a^{14} + \frac{11}{84} a^{13} + \frac{5}{21} a^{12} - \frac{2}{7} a^{11} - \frac{1}{84} a^{10} - \frac{5}{28} a^{9} - \frac{4}{21} a^{8} - \frac{1}{2} a^{7} - \frac{25}{84} a^{6} - \frac{1}{84} a^{5} - \frac{23}{84} a^{4} + \frac{31}{84} a^{3} - \frac{1}{2} a^{2} - \frac{5}{14} a - \frac{5}{21}$, $\frac{1}{973184588389923756} a^{15} - \frac{122417041844942}{81098715699160313} a^{14} - \frac{100343042115389015}{973184588389923756} a^{13} + \frac{59431326094805213}{243296147097480939} a^{12} + \frac{259087099174290011}{973184588389923756} a^{11} + \frac{10634456373111896}{34756592442497277} a^{10} - \frac{8203282031602309}{973184588389923756} a^{9} - \frac{66618862862233373}{486592294194961878} a^{8} + \frac{48670197646156565}{973184588389923756} a^{7} - \frac{221434471389235561}{486592294194961878} a^{6} - \frac{78281961607493321}{162197431398320626} a^{5} - \frac{25600343585520526}{243296147097480939} a^{4} + \frac{262307900547920119}{973184588389923756} a^{3} - \frac{13741916241961256}{81098715699160313} a^{2} - \frac{122804695767755941}{486592294194961878} a + \frac{89396844025502935}{243296147097480939}$

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:  \( \frac{322705973}{194286502542} a^{15} - \frac{100105245}{129524335028} a^{14} - \frac{2589155285}{388573005084} a^{13} - \frac{1651983655}{97143251271} a^{12} + \frac{6777213715}{194286502542} a^{11} + \frac{49316789795}{388573005084} a^{10} - \frac{51856458355}{388573005084} a^{9} - \frac{34507087135}{97143251271} a^{8} + \frac{747909095}{97143251271} a^{7} + \frac{559935615775}{388573005084} a^{6} + \frac{54564691799}{129524335028} a^{5} - \frac{1671484889215}{388573005084} a^{4} + \frac{439356397915}{388573005084} a^{3} + \frac{279418715055}{64762167514} a^{2} - \frac{295480479595}{194286502542} a - \frac{46744172558}{97143251271} \) (order $6$)
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:  \( 7830.84984215 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), 4.2.2000.1, 4.2.18000.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.324000000.4

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$