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Properties

Label 16.0.19680337591...1329.4
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 43^{8}$
Root discriminant $78.23$
Ramified primes $17, 43$
Class number $356$ (GRH)
Class group $[2, 178]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1543193, 427097, 2307490, 432383, 476036, 121358, 927, -4630, 14622, -5213, 2100, -318, 19, 12, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 12*x^14 + 12*x^13 + 19*x^12 - 318*x^11 + 2100*x^10 - 5213*x^9 + 14622*x^8 - 4630*x^7 + 927*x^6 + 121358*x^5 + 476036*x^4 + 432383*x^3 + 2307490*x^2 + 427097*x + 1543193)
 
gp: K = bnfinit(x^16 - 6*x^15 + 12*x^14 + 12*x^13 + 19*x^12 - 318*x^11 + 2100*x^10 - 5213*x^9 + 14622*x^8 - 4630*x^7 + 927*x^6 + 121358*x^5 + 476036*x^4 + 432383*x^3 + 2307490*x^2 + 427097*x + 1543193, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 12 x^{14} + 12 x^{13} + 19 x^{12} - 318 x^{11} + 2100 x^{10} - 5213 x^{9} + 14622 x^{8} - 4630 x^{7} + 927 x^{6} + 121358 x^{5} + 476036 x^{4} + 432383 x^{3} + 2307490 x^{2} + 427097 x + 1543193 \)

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:  \(1968033759133442969054057291329=17^{14}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.23$
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:  $17, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2314} a^{14} + \frac{75}{2314} a^{13} - \frac{40}{1157} a^{12} + \frac{749}{2314} a^{11} - \frac{155}{2314} a^{10} + \frac{336}{1157} a^{9} - \frac{1115}{2314} a^{8} + \frac{653}{2314} a^{7} - \frac{297}{1157} a^{6} - \frac{213}{2314} a^{5} - \frac{1149}{2314} a^{4} - \frac{10}{89} a^{3} - \frac{467}{2314} a^{2} - \frac{163}{2314} a + \frac{86}{1157}$, $\frac{1}{23112265684574802806748869756894906340838334} a^{15} - \frac{1333623898120073905743037045220003615423}{23112265684574802806748869756894906340838334} a^{14} - \frac{342349195813573918306004294274601162929457}{11556132842287401403374434878447453170419167} a^{13} + \frac{2581736338236838401369417213816034415712008}{11556132842287401403374434878447453170419167} a^{12} - \frac{6377574655741178331707511499280440848824587}{23112265684574802806748869756894906340838334} a^{11} - \frac{3620996083917010498565581504556435682603302}{11556132842287401403374434878447453170419167} a^{10} + \frac{1942680182936362581676828492470554912227648}{11556132842287401403374434878447453170419167} a^{9} - \frac{7132957492599232948839975234720592365647357}{23112265684574802806748869756894906340838334} a^{8} - \frac{5755358855232805088240182246469152183443226}{11556132842287401403374434878447453170419167} a^{7} + \frac{1183799879450031435526901771565082693248322}{11556132842287401403374434878447453170419167} a^{6} + \frac{1151382467255204000738999673834794478953541}{23112265684574802806748869756894906340838334} a^{5} - \frac{359085497588468946735137622562464202657985}{11556132842287401403374434878447453170419167} a^{4} - \frac{4273631703180602540373296446671491032823323}{11556132842287401403374434878447453170419167} a^{3} + \frac{1916189129613163335607705864966314109307953}{23112265684574802806748869756894906340838334} a^{2} + \frac{16843567712308940882976122412342456419478}{888933295560569338721110375265188705416859} a + \frac{9363492039866256917770040665081533356501579}{23112265684574802806748869756894906340838334}$

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

Class group and class number

$C_{2}\times C_{178}$, which has order $356$ (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:  \( 535402.98004 \) (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

\(\Q(\sqrt{17}) \), 4.2.12427.1, 4.4.4913.1, 4.2.211259.1, 8.0.1402866265591073.2, 8.4.758716206377.1, 8.4.44630365081.1

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
$43$43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$