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Properties

Label 16.0.62086835430...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 11^{6}\cdot 37^{2}$
Root discriminant $17.26$
Ramified primes $2, 5, 11, 37$
Class number $1$
Class group Trivial
Galois group 16T1772

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 66, 154, 176, 167, 150, 170, 120, 87, 76, 36, 26, 5, 4, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 + 4*x^13 + 5*x^12 + 26*x^11 + 36*x^10 + 76*x^9 + 87*x^8 + 120*x^7 + 170*x^6 + 150*x^5 + 167*x^4 + 176*x^3 + 154*x^2 + 66*x + 11)
 
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 + 4*x^13 + 5*x^12 + 26*x^11 + 36*x^10 + 76*x^9 + 87*x^8 + 120*x^7 + 170*x^6 + 150*x^5 + 167*x^4 + 176*x^3 + 154*x^2 + 66*x + 11, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 4 x^{14} + 4 x^{13} + 5 x^{12} + 26 x^{11} + 36 x^{10} + 76 x^{9} + 87 x^{8} + 120 x^{7} + 170 x^{6} + 150 x^{5} + 167 x^{4} + 176 x^{3} + 154 x^{2} + 66 x + 11 \)

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:  \(62086835430400000000=2^{16}\cdot 5^{8}\cdot 11^{6}\cdot 37^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.26$
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:  $2, 5, 11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{3}$, $\frac{1}{50} a^{14} - \frac{3}{25} a^{13} - \frac{3}{25} a^{12} + \frac{7}{50} a^{11} + \frac{3}{25} a^{10} - \frac{11}{50} a^{9} + \frac{1}{50} a^{8} + \frac{21}{50} a^{7} + \frac{19}{50} a^{6} + \frac{1}{10} a^{5} + \frac{2}{25} a^{4} - \frac{11}{50} a^{3} - \frac{1}{2} a^{2} - \frac{21}{50}$, $\frac{1}{481607248950} a^{15} + \frac{3655110983}{481607248950} a^{14} + \frac{7421545979}{32107149930} a^{13} - \frac{13884356476}{240803624475} a^{12} + \frac{1616296931}{53511916550} a^{11} + \frac{23558169424}{240803624475} a^{10} - \frac{97021724803}{481607248950} a^{9} - \frac{2308008877}{10702383310} a^{8} + \frac{25913966621}{160535749650} a^{7} + \frac{21044641769}{53511916550} a^{6} + \frac{15624350662}{240803624475} a^{5} - \frac{21242750338}{48160724895} a^{4} - \frac{61892036677}{240803624475} a^{3} - \frac{1022194167}{2140476662} a^{2} - \frac{5020574773}{240803624475} a - \frac{120612623219}{481607248950}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 2048.58767266 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1772:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 148 conjugacy class representatives for t16n1772 are not computed
Character table for t16n1772 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.0.212960000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.8.8.10$x^{8} + 2 x^{6} + 8 x^{3} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.8.10$x^{8} + 2 x^{6} + 8 x^{3} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$