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Properties

Label 16.0.42679667686...5801.2
Degree $16$
Signature $[0, 8]$
Discriminant $23^{12}\cdot 41^{7}$
Root discriminant $53.32$
Ramified primes $23, 41$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2065061, -3791354, 1643530, 1717890, -1637207, -91475, 483073, -95606, -64784, 26164, 3213, -2943, 184, 165, -26, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 26*x^14 + 165*x^13 + 184*x^12 - 2943*x^11 + 3213*x^10 + 26164*x^9 - 64784*x^8 - 95606*x^7 + 483073*x^6 - 91475*x^5 - 1637207*x^4 + 1717890*x^3 + 1643530*x^2 - 3791354*x + 2065061)
 
gp: K = bnfinit(x^16 - 3*x^15 - 26*x^14 + 165*x^13 + 184*x^12 - 2943*x^11 + 3213*x^10 + 26164*x^9 - 64784*x^8 - 95606*x^7 + 483073*x^6 - 91475*x^5 - 1637207*x^4 + 1717890*x^3 + 1643530*x^2 - 3791354*x + 2065061, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 26 x^{14} + 165 x^{13} + 184 x^{12} - 2943 x^{11} + 3213 x^{10} + 26164 x^{9} - 64784 x^{8} - 95606 x^{7} + 483073 x^{6} - 91475 x^{5} - 1637207 x^{4} + 1717890 x^{3} + 1643530 x^{2} - 3791354 x + 2065061 \)

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:  \(4267966768632939662191535801=23^{12}\cdot 41^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.32$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{59} a^{13} + \frac{16}{59} a^{12} + \frac{6}{59} a^{11} - \frac{13}{59} a^{10} + \frac{17}{59} a^{9} + \frac{24}{59} a^{8} + \frac{14}{59} a^{7} + \frac{9}{59} a^{6} - \frac{9}{59} a^{5} - \frac{26}{59} a^{4} + \frac{7}{59} a^{3} + \frac{22}{59} a^{2} + \frac{23}{59} a + \frac{27}{59}$, $\frac{1}{59} a^{14} - \frac{14}{59} a^{12} + \frac{9}{59} a^{11} - \frac{11}{59} a^{10} - \frac{12}{59} a^{9} - \frac{16}{59} a^{8} + \frac{21}{59} a^{7} + \frac{24}{59} a^{6} + \frac{10}{59} a^{4} + \frac{28}{59} a^{3} + \frac{25}{59} a^{2} + \frac{13}{59} a - \frac{19}{59}$, $\frac{1}{1049375208215492723761909368479337011} a^{15} - \frac{1128267425700953991047249161726022}{1049375208215492723761909368479337011} a^{14} + \frac{5221136777329201711153825964672912}{1049375208215492723761909368479337011} a^{13} - \frac{66521148381173768129393215917540912}{1049375208215492723761909368479337011} a^{12} - \frac{76383336736376396338104473404872184}{1049375208215492723761909368479337011} a^{11} + \frac{112192400349287158230916081369954148}{1049375208215492723761909368479337011} a^{10} - \frac{502373985996084598638737176434882097}{1049375208215492723761909368479337011} a^{9} - \frac{499286628705450401769312163165233830}{1049375208215492723761909368479337011} a^{8} + \frac{385177951628035183227307563993069286}{1049375208215492723761909368479337011} a^{7} + \frac{332151941929641932403975050016366030}{1049375208215492723761909368479337011} a^{6} - \frac{8136089230486883687080534966925777}{22327132089691334548125731244241213} a^{5} + \frac{409466297505834001855452706050903224}{1049375208215492723761909368479337011} a^{4} - \frac{281158301630977802160541086786005583}{1049375208215492723761909368479337011} a^{3} - \frac{195381389217702340916749136666375732}{1049375208215492723761909368479337011} a^{2} + \frac{502517446119274396649634924252010834}{1049375208215492723761909368479337011} a + \frac{250733679618226474106627316701819066}{1049375208215492723761909368479337011}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 6935697.82772 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-23}) \), 4.0.21689.1, 8.0.19286921561.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ $16$ $16$ R ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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$23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$