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Properties

Label 16.4.56148868668...6129.7
Degree $16$
Signature $[4, 6]$
Discriminant $23^{6}\cdot 41^{14}$
Root discriminant $83.53$
Ramified primes $23, 41$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-212671, 296055, 586484, 662650, 264039, 395156, 39783, 166391, -21833, 9261, -11210, 1237, -312, 147, -1, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - x^14 + 147*x^13 - 312*x^12 + 1237*x^11 - 11210*x^10 + 9261*x^9 - 21833*x^8 + 166391*x^7 + 39783*x^6 + 395156*x^5 + 264039*x^4 + 662650*x^3 + 586484*x^2 + 296055*x - 212671)
 
gp: K = bnfinit(x^16 - 8*x^15 - x^14 + 147*x^13 - 312*x^12 + 1237*x^11 - 11210*x^10 + 9261*x^9 - 21833*x^8 + 166391*x^7 + 39783*x^6 + 395156*x^5 + 264039*x^4 + 662650*x^3 + 586484*x^2 + 296055*x - 212671, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - x^{14} + 147 x^{13} - 312 x^{12} + 1237 x^{11} - 11210 x^{10} + 9261 x^{9} - 21833 x^{8} + 166391 x^{7} + 39783 x^{6} + 395156 x^{5} + 264039 x^{4} + 662650 x^{3} + 586484 x^{2} + 296055 x - 212671 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5614886866882301027209678756129=23^{6}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.53$
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:  $23, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4}$, $\frac{1}{20} a^{13} + \frac{1}{20} a^{12} + \frac{3}{20} a^{11} - \frac{1}{4} a^{10} - \frac{1}{20} a^{9} + \frac{1}{20} a^{8} - \frac{1}{20} a^{7} + \frac{1}{20} a^{6} + \frac{1}{10} a^{5} - \frac{1}{10} a^{4} - \frac{1}{5} a^{3} + \frac{1}{10} a^{2} - \frac{1}{4} a - \frac{9}{20}$, $\frac{1}{20} a^{14} + \frac{1}{10} a^{12} + \frac{1}{10} a^{11} + \frac{1}{5} a^{10} + \frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{1}{10} a^{7} + \frac{1}{20} a^{6} + \frac{3}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{7}{20} a^{2} + \frac{3}{10} a - \frac{1}{20}$, $\frac{1}{5084829827112976071450459371120027148947096620} a^{15} - \frac{21595422862160974007977310788942104297983993}{1016965965422595214290091874224005429789419324} a^{14} - \frac{2523401886707832935981932568958076003809952}{254241491355648803572522968556001357447354831} a^{13} - \frac{44304199972008476062537921858463680474613127}{1016965965422595214290091874224005429789419324} a^{12} + \frac{71453581634145692892266334424227075491954167}{1271207456778244017862614842780006787236774155} a^{11} - \frac{597094853044853025792575291504147336674896443}{5084829827112976071450459371120027148947096620} a^{10} - \frac{178960585464748635140278436395645907116048}{6871391658260778474933053204216252903982563} a^{9} - \frac{198093117898138456453108514569381135117709869}{1016965965422595214290091874224005429789419324} a^{8} - \frac{249834237848489237300687683838472291806468857}{5084829827112976071450459371120027148947096620} a^{7} - \frac{474439252110895213030955796860435206804140953}{2542414913556488035725229685560013574473548310} a^{6} - \frac{126535011501687113616204352554882551998416403}{2542414913556488035725229685560013574473548310} a^{5} - \frac{19282599782992009345609515535136546114647379}{508482982711297607145045937112002714894709662} a^{4} + \frac{40391492191515476093723564643015124832021111}{164026768616547615208079334552258940288616020} a^{3} + \frac{1726706142517584589631394547771669945845254187}{5084829827112976071450459371120027148947096620} a^{2} + \frac{848239030354539303620750514974074655290531169}{5084829827112976071450459371120027148947096620} a + \frac{1018383881005150148567740177652818801417756059}{2542414913556488035725229685560013574473548310}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  $9$
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:  \( 191680973.532 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 4.2.38663.1, 4.2.1585183.1, 8.4.103025010883049.2 x2, 8.4.2512805143489.1

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 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$