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Properties

Label 16.8.99253389787...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 11^{4}\cdot 19^{2}\cdot 29^{8}\cdot 31^{2}$
Root discriminant $48.67$
Ramified primes $5, 11, 19, 29, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T860

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89899, -30821, -77480, 63447, -5460, -31296, 12482, -576, 760, 1596, -512, 184, -118, 9, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 2*x^14 + 9*x^13 - 118*x^12 + 184*x^11 - 512*x^10 + 1596*x^9 + 760*x^8 - 576*x^7 + 12482*x^6 - 31296*x^5 - 5460*x^4 + 63447*x^3 - 77480*x^2 - 30821*x + 89899)
 
gp: K = bnfinit(x^16 - 2*x^15 - 2*x^14 + 9*x^13 - 118*x^12 + 184*x^11 - 512*x^10 + 1596*x^9 + 760*x^8 - 576*x^7 + 12482*x^6 - 31296*x^5 - 5460*x^4 + 63447*x^3 - 77480*x^2 - 30821*x + 89899, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 2 x^{14} + 9 x^{13} - 118 x^{12} + 184 x^{11} - 512 x^{10} + 1596 x^{9} + 760 x^{8} - 576 x^{7} + 12482 x^{6} - 31296 x^{5} - 5460 x^{4} + 63447 x^{3} - 77480 x^{2} - 30821 x + 89899 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(992533897870849042547265625=5^{8}\cdot 11^{4}\cdot 19^{2}\cdot 29^{8}\cdot 31^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.67$
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:  $5, 11, 19, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2071} a^{13} - \frac{755}{2071} a^{12} + \frac{27}{109} a^{11} + \frac{149}{2071} a^{10} + \frac{1006}{2071} a^{9} - \frac{85}{2071} a^{8} + \frac{329}{2071} a^{7} - \frac{878}{2071} a^{6} - \frac{387}{2071} a^{5} + \frac{311}{2071} a^{4} + \frac{636}{2071} a^{3} - \frac{26}{2071} a^{2} - \frac{764}{2071} a - \frac{333}{2071}$, $\frac{1}{36426819} a^{14} - \frac{7945}{36426819} a^{13} + \frac{34531}{334191} a^{12} + \frac{554381}{12142273} a^{11} - \frac{215058}{1103843} a^{10} - \frac{14506577}{36426819} a^{9} - \frac{9532279}{36426819} a^{8} - \frac{74301}{12142273} a^{7} + \frac{1509077}{12142273} a^{6} - \frac{4756591}{12142273} a^{5} - \frac{13325659}{36426819} a^{4} - \frac{7869898}{36426819} a^{3} - \frac{6059960}{36426819} a^{2} + \frac{548303}{12142273} a - \frac{5976712}{36426819}$, $\frac{1}{447520832999415499418913819} a^{15} - \frac{1940111867475781537}{447520832999415499418913819} a^{14} + \frac{13597796175943068929848}{447520832999415499418913819} a^{13} - \frac{28396408851922475934751408}{149173610999805166472971273} a^{12} + \frac{19481547177928375813692156}{149173610999805166472971273} a^{11} + \frac{164084734334129601685976242}{447520832999415499418913819} a^{10} - \frac{157341504238576996583674294}{447520832999415499418913819} a^{9} + \frac{50481324070617422105156784}{149173610999805166472971273} a^{8} - \frac{1459998094063681713895923}{149173610999805166472971273} a^{7} - \frac{41235642900447005899186937}{149173610999805166472971273} a^{6} - \frac{71483531571857101829730223}{447520832999415499418913819} a^{5} + \frac{3211268269584235583441879}{447520832999415499418913819} a^{4} - \frac{203253544647503940087220586}{447520832999415499418913819} a^{3} - \frac{57170201757991323392721149}{149173610999805166472971273} a^{2} - \frac{147652104171241693335565987}{447520832999415499418913819} a + \frac{48604030423404049492836484}{149173610999805166472971273}$

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:  $11$
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:  \( 13039180.396 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T860:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n860 are not computed
Character table for t16n860 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{145}) \), 4.4.725.1 x2, 4.4.4205.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
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$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$