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Properties

Label 16.8.32454990969...0912.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{67}\cdot 17^{6}\cdot 977^{4}$
Root discriminant $294.75$
Ramified primes $2, 17, 977$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T937

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8426089658305568, 0, -961880064552192, 0, 20442712327776, 0, 454984179136, 0, -5052241000, 0, -34006720, 0, 100792, 0, 776, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 776*x^14 + 100792*x^12 - 34006720*x^10 - 5052241000*x^8 + 454984179136*x^6 + 20442712327776*x^4 - 961880064552192*x^2 + 8426089658305568)
 
gp: K = bnfinit(x^16 + 776*x^14 + 100792*x^12 - 34006720*x^10 - 5052241000*x^8 + 454984179136*x^6 + 20442712327776*x^4 - 961880064552192*x^2 + 8426089658305568, 1)
 

Normalized defining polynomial

\( x^{16} + 776 x^{14} + 100792 x^{12} - 34006720 x^{10} - 5052241000 x^{8} + 454984179136 x^{6} + 20442712327776 x^{4} - 961880064552192 x^{2} + 8426089658305568 \)

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:  \(3245499096990365116593765338925510950912=2^{67}\cdot 17^{6}\cdot 977^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $294.75$
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:  $2, 17, 977$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{7816} a^{10} + \frac{97}{977} a^{8} - \frac{102}{977} a^{6} + \frac{87}{977} a^{4} - \frac{256}{977} a^{2}$, $\frac{1}{7816} a^{11} + \frac{97}{977} a^{9} - \frac{102}{977} a^{7} + \frac{87}{977} a^{5} - \frac{256}{977} a^{3}$, $\frac{1}{7636232} a^{12} - \frac{201}{7636232} a^{10} - \frac{82170}{954529} a^{8} + \frac{288389}{1909058} a^{6} - \frac{388381}{1909058} a^{4} - \frac{439}{977} a^{2}$, $\frac{1}{15272464} a^{13} + \frac{97}{1909058} a^{11} + \frac{12599}{1909058} a^{9} + \frac{89081}{3818116} a^{7} + \frac{368073}{1909058} a^{5} + \frac{141}{977} a^{3}$, $\frac{1}{66050914037041598758799006428862638057338030448} a^{14} + \frac{45469046056662804228358024297338890002}{4128182127315099922424937901803914878583626903} a^{12} + \frac{595292735684256355361105879405549767222783}{33025457018520799379399503214431319028669015224} a^{10} - \frac{233943213921415341733248388089321803727994177}{4128182127315099922424937901803914878583626903} a^{8} + \frac{501280266823860761173107965262176102341128009}{4128182127315099922424937901803914878583626903} a^{6} - \frac{1028933904765507003668892772504377270334035}{4225365534611156522441082806349964051774439} a^{4} - \frac{275466356348199840256432462548381947742}{617833825794875935435163445876584888401} a^{2} - \frac{68407677558285024368019128963585250}{260391160248307035224645922158835223}$, $\frac{1}{66050914037041598758799006428862638057338030448} a^{15} + \frac{45469046056662804228358024297338890002}{4128182127315099922424937901803914878583626903} a^{13} + \frac{595292735684256355361105879405549767222783}{33025457018520799379399503214431319028669015224} a^{11} - \frac{233943213921415341733248388089321803727994177}{4128182127315099922424937901803914878583626903} a^{9} + \frac{501280266823860761173107965262176102341128009}{4128182127315099922424937901803914878583626903} a^{7} - \frac{1028933904765507003668892772504377270334035}{4225365534611156522441082806349964051774439} a^{5} - \frac{275466356348199840256432462548381947742}{617833825794875935435163445876584888401} a^{3} - \frac{68407677558285024368019128963585250}{260391160248307035224645922158835223} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, 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:  $11$
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:  \( 19271757965800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T937:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n937
Character table for t16n937 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.4352.1, 8.8.9697230848.1

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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
977Data not computed