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Properties

Label 16.8.48774787192...5504.5
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 3^{12}\cdot 17^{6}\cdot 97^{4}$
Root discriminant $82.80$
Ramified primes $2, 3, 17, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![123926857, -247607232, 91050580, 33646688, -11490419, -4482264, -1696040, 1537644, 203518, -151076, 9172, 1496, -1540, 248, 28, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 28*x^14 + 248*x^13 - 1540*x^12 + 1496*x^11 + 9172*x^10 - 151076*x^9 + 203518*x^8 + 1537644*x^7 - 1696040*x^6 - 4482264*x^5 - 11490419*x^4 + 33646688*x^3 + 91050580*x^2 - 247607232*x + 123926857)
 
gp: K = bnfinit(x^16 - 4*x^15 + 28*x^14 + 248*x^13 - 1540*x^12 + 1496*x^11 + 9172*x^10 - 151076*x^9 + 203518*x^8 + 1537644*x^7 - 1696040*x^6 - 4482264*x^5 - 11490419*x^4 + 33646688*x^3 + 91050580*x^2 - 247607232*x + 123926857, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 28 x^{14} + 248 x^{13} - 1540 x^{12} + 1496 x^{11} + 9172 x^{10} - 151076 x^{9} + 203518 x^{8} + 1537644 x^{7} - 1696040 x^{6} - 4482264 x^{5} - 11490419 x^{4} + 33646688 x^{3} + 91050580 x^{2} - 247607232 x + 123926857 \)

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:  \(4877478719269533554834097045504=2^{32}\cdot 3^{12}\cdot 17^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.80$
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:  $2, 3, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{11} - \frac{1}{5} a^{9} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{831781307201052925873895766154953744013613969952231541420385} a^{15} + \frac{73058137103689007733062180847314614781906236790034823620}{166356261440210585174779153230990748802722793990446308284077} a^{14} + \frac{7586726475712510972827461953776374934813380099729717800134}{166356261440210585174779153230990748802722793990446308284077} a^{13} + \frac{79393013328629275241210443822442172281690454201429391235787}{831781307201052925873895766154953744013613969952231541420385} a^{12} + \frac{50979186414515600843813780477129558196815527904654633029011}{831781307201052925873895766154953744013613969952231541420385} a^{11} - \frac{60750902411918025930977509907777065937930626892296359875041}{166356261440210585174779153230990748802722793990446308284077} a^{10} + \frac{288324790632674552897174016719448169177297713046152418107619}{831781307201052925873895766154953744013613969952231541420385} a^{9} - \frac{24679397107848695086972033625379272228209750165834986693551}{831781307201052925873895766154953744013613969952231541420385} a^{8} - \frac{69679978254588110066931800936356010793701083865622242338789}{166356261440210585174779153230990748802722793990446308284077} a^{7} - \frac{213994257503619122492448560070083817973416790227362689025251}{831781307201052925873895766154953744013613969952231541420385} a^{6} - \frac{314165837471433582855815391527147512999621404415319831470959}{831781307201052925873895766154953744013613969952231541420385} a^{5} - \frac{217587950824293673274181070655479627653309967630471555176442}{831781307201052925873895766154953744013613969952231541420385} a^{4} - \frac{214590986521864685846045334493010823761711031487091072937299}{831781307201052925873895766154953744013613969952231541420385} a^{3} + \frac{333784460202164569554365659614315933557181239876298490631899}{831781307201052925873895766154953744013613969952231541420385} a^{2} - \frac{216474440733602517569535854936450372255353694626714346290838}{831781307201052925873895766154953744013613969952231541420385} a - \frac{183423181340495333780386832459167848819292813322687984841299}{831781307201052925873895766154953744013613969952231541420385}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 516599699.875 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), 4.4.4352.1, 4.4.9792.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
3Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
97Data not computed