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Properties

Label 16.8.17555500852...1584.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{56}\cdot 3^{8}\cdot 41^{2}\cdot 47^{2}$
Root discriminant $50.44$
Ramified primes $2, 3, 41, 47$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-83234, 154224, -74208, 73184, -69980, -11680, 32984, -16512, 4942, 440, -1312, 840, -232, -16, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 16*x^13 - 232*x^12 + 840*x^11 - 1312*x^10 + 440*x^9 + 4942*x^8 - 16512*x^7 + 32984*x^6 - 11680*x^5 - 69980*x^4 + 73184*x^3 - 74208*x^2 + 154224*x - 83234)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 16*x^13 - 232*x^12 + 840*x^11 - 1312*x^10 + 440*x^9 + 4942*x^8 - 16512*x^7 + 32984*x^6 - 11680*x^5 - 69980*x^4 + 73184*x^3 - 74208*x^2 + 154224*x - 83234, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 16 x^{13} - 232 x^{12} + 840 x^{11} - 1312 x^{10} + 440 x^{9} + 4942 x^{8} - 16512 x^{7} + 32984 x^{6} - 11680 x^{5} - 69980 x^{4} + 73184 x^{3} - 74208 x^{2} + 154224 x - 83234 \)

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:  \(1755550085243509039467331584=2^{56}\cdot 3^{8}\cdot 41^{2}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.44$
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:  $2, 3, 41, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{647961881} a^{14} - \frac{297829948}{647961881} a^{13} + \frac{84783914}{647961881} a^{12} + \frac{32853483}{92565983} a^{11} + \frac{3361783}{92565983} a^{10} + \frac{42369215}{92565983} a^{9} + \frac{213698741}{647961881} a^{8} - \frac{173801042}{647961881} a^{7} - \frac{256948935}{647961881} a^{6} - \frac{251105353}{647961881} a^{5} + \frac{130929998}{647961881} a^{4} + \frac{137344436}{647961881} a^{3} + \frac{192257532}{647961881} a^{2} + \frac{144724640}{647961881} a + \frac{90371637}{647961881}$, $\frac{1}{4379030030247498041098878604423} a^{15} - \frac{1238521785342628027819}{4379030030247498041098878604423} a^{14} + \frac{1025814300623894694163254640715}{4379030030247498041098878604423} a^{13} + \frac{91822426912631119107086725694}{4379030030247498041098878604423} a^{12} + \frac{70737324555067159953777507934}{625575718606785434442696943489} a^{11} + \frac{2139179461791271849313947389}{7918679982364372587882239791} a^{10} + \frac{923658231990443402686706148008}{4379030030247498041098878604423} a^{9} - \frac{1454182660951060909164679629917}{4379030030247498041098878604423} a^{8} + \frac{172268099697159754215939928011}{4379030030247498041098878604423} a^{7} + \frac{244160822926569934850310653741}{4379030030247498041098878604423} a^{6} - \frac{2135400119322454375834981526613}{4379030030247498041098878604423} a^{5} + \frac{678499770248364718624535934735}{4379030030247498041098878604423} a^{4} - \frac{2876374877737267956018372760}{7918679982364372587882239791} a^{3} - \frac{785817125531130043229213646277}{4379030030247498041098878604423} a^{2} + \frac{1697683437924085480487742384956}{4379030030247498041098878604423} a + \frac{298496986099480885174751782840}{4379030030247498041098878604423}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 30869807.4108 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n797 are not computed
Character table for t16n797 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\zeta_{16})^+\), 4.4.18432.1, \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{48})^+\)

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
47Data not computed