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Properties

Label 16.0.17050453937...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 11^{8}\cdot 19^{4}$
Root discriminant $28.31$
Ramified primes $5, 11, 19$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2\times C_2^3.C_4$ (as 16T99)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, -63, 185, 1983, 2150, 264, -170, 345, 180, -31, 65, 68, -5, -13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 13*x^14 - 5*x^13 + 68*x^12 + 65*x^11 - 31*x^10 + 180*x^9 + 345*x^8 - 170*x^7 + 264*x^6 + 2150*x^5 + 1983*x^4 + 185*x^3 - 63*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^16 - 13*x^14 - 5*x^13 + 68*x^12 + 65*x^11 - 31*x^10 + 180*x^9 + 345*x^8 - 170*x^7 + 264*x^6 + 2150*x^5 + 1983*x^4 + 185*x^3 - 63*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 13 x^{14} - 5 x^{13} + 68 x^{12} + 65 x^{11} - 31 x^{10} + 180 x^{9} + 345 x^{8} - 170 x^{7} + 264 x^{6} + 2150 x^{5} + 1983 x^{4} + 185 x^{3} - 63 x^{2} - 5 x + 1 \)

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:  \(170504539372564697265625=5^{14}\cdot 11^{8}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.31$
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:  $5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{29} a^{13} + \frac{11}{29} a^{12} - \frac{5}{29} a^{11} + \frac{7}{29} a^{10} + \frac{11}{29} a^{9} + \frac{8}{29} a^{8} + \frac{9}{29} a^{7} + \frac{10}{29} a^{6} - \frac{8}{29} a^{4} - \frac{6}{29} a^{3} + \frac{1}{29} a^{2} - \frac{7}{29} a - \frac{6}{29}$, $\frac{1}{29} a^{14} - \frac{10}{29} a^{12} + \frac{4}{29} a^{11} - \frac{8}{29} a^{10} + \frac{3}{29} a^{9} + \frac{8}{29} a^{8} - \frac{2}{29} a^{7} + \frac{6}{29} a^{6} - \frac{8}{29} a^{5} - \frac{5}{29} a^{4} + \frac{9}{29} a^{3} + \frac{11}{29} a^{2} + \frac{13}{29} a + \frac{8}{29}$, $\frac{1}{66497195742034019} a^{15} + \frac{802807708461778}{66497195742034019} a^{14} - \frac{939748799377787}{66497195742034019} a^{13} + \frac{15591964514535772}{66497195742034019} a^{12} + \frac{19284238713275723}{66497195742034019} a^{11} + \frac{29904720339390432}{66497195742034019} a^{10} - \frac{544913065306420}{2293006749725311} a^{9} + \frac{3232725382127127}{66497195742034019} a^{8} + \frac{24443274643626690}{66497195742034019} a^{7} - \frac{19259206939575245}{66497195742034019} a^{6} + \frac{267669554763832}{936580221718789} a^{5} + \frac{41851620333992}{100600901273879} a^{4} + \frac{41576882293529}{100600901273879} a^{3} - \frac{338620539804503}{936580221718789} a^{2} - \frac{20666268808535809}{66497195742034019} a - \frac{26906993557812656}{66497195742034019}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{112421368582983}{66497195742034019} a^{15} + \frac{2370973133022157}{66497195742034019} a^{14} - \frac{2702767599861699}{66497195742034019} a^{13} - \frac{29540442169001051}{66497195742034019} a^{12} + \frac{9078605567972761}{66497195742034019} a^{11} + \frac{155266616151473579}{66497195742034019} a^{10} + \frac{86946983392222411}{66497195742034019} a^{9} - \frac{41989329083835604}{66497195742034019} a^{8} + \frac{483227302144483537}{66497195742034019} a^{7} + \frac{558637924033980074}{66497195742034019} a^{6} - \frac{6306470121771189}{936580221718789} a^{5} + \frac{1753818212461350}{100600901273879} a^{4} + \frac{6939027606462156}{100600901273879} a^{3} + \frac{44564264076515416}{936580221718789} a^{2} + \frac{130461284279982025}{66497195742034019} a - \frac{3779322285196040}{66497195742034019} \) (order $10$)
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:  \( 55211.8655101 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^3.C_4$ (as 16T99):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2\times C_2^3.C_4$
Character table for $C_2\times C_2^3.C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-11}) \), \(\Q(\zeta_{5})\), 4.4.15125.1, \(\Q(\sqrt{5}, \sqrt{-11})\), 8.4.412921953125.1, 8.4.412921953125.2, 8.0.228765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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}$