↑

Properties

Label 16.0.92600652331...0625.7
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{14}$
Root discriminant $86.18$
Ramified primes $5, 41$
Class number $600$ (GRH)
Class group $[2, 300]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![155285056, -73814496, -620444, -18892900, 43579355, -26273899, 8594450, -1759715, 211046, -63735, 16060, -4449, 1270, -105, 46, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 46*x^14 - 105*x^13 + 1270*x^12 - 4449*x^11 + 16060*x^10 - 63735*x^9 + 211046*x^8 - 1759715*x^7 + 8594450*x^6 - 26273899*x^5 + 43579355*x^4 - 18892900*x^3 - 620444*x^2 - 73814496*x + 155285056)
 
gp: K = bnfinit(x^16 - x^15 + 46*x^14 - 105*x^13 + 1270*x^12 - 4449*x^11 + 16060*x^10 - 63735*x^9 + 211046*x^8 - 1759715*x^7 + 8594450*x^6 - 26273899*x^5 + 43579355*x^4 - 18892900*x^3 - 620444*x^2 - 73814496*x + 155285056, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 46 x^{14} - 105 x^{13} + 1270 x^{12} - 4449 x^{11} + 16060 x^{10} - 63735 x^{9} + 211046 x^{8} - 1759715 x^{7} + 8594450 x^{6} - 26273899 x^{5} + 43579355 x^{4} - 18892900 x^{3} - 620444 x^{2} - 73814496 x + 155285056 \)

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:  \(9260065233133681348183837890625=5^{12}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.18$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{3}{16} a^{5} - \frac{3}{16} a^{4} + \frac{5}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{9} - \frac{1}{4} a^{7} + \frac{3}{16} a^{6} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{13} - \frac{1}{32} a^{12} + \frac{1}{64} a^{11} - \frac{1}{64} a^{9} - \frac{1}{16} a^{8} + \frac{15}{64} a^{7} - \frac{3}{32} a^{6} + \frac{11}{64} a^{5} - \frac{11}{64} a^{3} + \frac{5}{16} a^{2} - \frac{5}{16} a - \frac{1}{4}$, $\frac{1}{128} a^{14} - \frac{3}{128} a^{12} - \frac{1}{64} a^{11} - \frac{5}{128} a^{10} + \frac{3}{64} a^{9} - \frac{9}{128} a^{8} + \frac{1}{16} a^{7} - \frac{1}{128} a^{6} - \frac{11}{64} a^{5} + \frac{9}{128} a^{4} + \frac{1}{64} a^{3} + \frac{9}{32} a^{2} + \frac{3}{16} a + \frac{1}{4}$, $\frac{1}{1039736457045233179896183917393199158265972394404033664} a^{15} - \frac{540819804738278868421618111075821198976320335025159}{259934114261308294974045979348299789566493098601008416} a^{14} - \frac{6257900245715157662611975508820172373162475745700303}{1039736457045233179896183917393199158265972394404033664} a^{13} + \frac{13104734695790854405191219094687191478152834653238441}{519868228522616589948091958696599579132986197202016832} a^{12} - \frac{24677953441340788050765043786714080513374928228266041}{1039736457045233179896183917393199158265972394404033664} a^{11} - \frac{10365632404972116483287365459800759076801512766112367}{519868228522616589948091958696599579132986197202016832} a^{10} + \frac{78629727706284299079764066252626350233837007589935203}{1039736457045233179896183917393199158265972394404033664} a^{9} + \frac{15663223276827720655012692906394574994283468929981585}{259934114261308294974045979348299789566493098601008416} a^{8} - \frac{29457312261038230873290172232421887544891191705698133}{1039736457045233179896183917393199158265972394404033664} a^{7} - \frac{4280219760655197772441907314578807242923881417052669}{519868228522616589948091958696599579132986197202016832} a^{6} + \frac{78050844692729992584280296610720463119588536628768781}{1039736457045233179896183917393199158265972394404033664} a^{5} - \frac{46313917347856143690782114511783533652164025383608677}{519868228522616589948091958696599579132986197202016832} a^{4} - \frac{9451952668870356693584022326340785227954264078456335}{129967057130654147487022989674149894783246549300504208} a^{3} - \frac{1737859487444112296377485985160554418324882253606971}{129967057130654147487022989674149894783246549300504208} a^{2} - \frac{2564141386571025291285361065906271934921219584620473}{64983528565327073743511494837074947391623274650252104} a - \frac{1399840310322351655711660183464090787934032499490101}{16245882141331768435877873709268736847905818662563026}$

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

Class group and class number

$C_{2}\times C_{300}$, which has order $600$ (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:  \( -1 \) (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:  \( 10205497.328 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.0.8405.1, 4.4.1723025.1, 4.0.344605.1, 8.8.3043035529390625.1, 8.0.3043035529390625.1, 8.0.2968815150625.3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed