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Properties

Label 16.0.10564402173...1889.2
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 89^{4}$
Root discriminant $36.64$
Ramified primes $17, 89$
Class number $140$ (GRH)
Class group $[140]$ (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![4096, 9216, 26112, 19840, 41808, 14640, 32800, 6850, 15141, 2113, 3660, 101, 473, -31, 34, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 34*x^14 - 31*x^13 + 473*x^12 + 101*x^11 + 3660*x^10 + 2113*x^9 + 15141*x^8 + 6850*x^7 + 32800*x^6 + 14640*x^5 + 41808*x^4 + 19840*x^3 + 26112*x^2 + 9216*x + 4096)
 
gp: K = bnfinit(x^16 - 3*x^15 + 34*x^14 - 31*x^13 + 473*x^12 + 101*x^11 + 3660*x^10 + 2113*x^9 + 15141*x^8 + 6850*x^7 + 32800*x^6 + 14640*x^5 + 41808*x^4 + 19840*x^3 + 26112*x^2 + 9216*x + 4096, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 34 x^{14} - 31 x^{13} + 473 x^{12} + 101 x^{11} + 3660 x^{10} + 2113 x^{9} + 15141 x^{8} + 6850 x^{7} + 32800 x^{6} + 14640 x^{5} + 41808 x^{4} + 19840 x^{3} + 26112 x^{2} + 9216 x + 4096 \)

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:  \(10564402173046133902941889=17^{14}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.64$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{8} + \frac{3}{8} a^{7} + \frac{3}{8} a^{6} + \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{9} - \frac{5}{16} a^{8} - \frac{5}{16} a^{7} + \frac{1}{8} a^{6} + \frac{5}{16} a^{5} - \frac{1}{16} a^{4}$, $\frac{1}{64} a^{13} + \frac{1}{64} a^{12} - \frac{1}{32} a^{11} + \frac{1}{64} a^{10} - \frac{3}{64} a^{9} + \frac{17}{64} a^{8} + \frac{3}{8} a^{7} + \frac{9}{64} a^{6} + \frac{25}{64} a^{5} - \frac{1}{32} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{3328} a^{14} + \frac{1}{3328} a^{13} + \frac{27}{1664} a^{12} + \frac{201}{3328} a^{11} - \frac{355}{3328} a^{10} - \frac{599}{3328} a^{9} + \frac{9}{52} a^{8} - \frac{1007}{3328} a^{7} + \frac{1193}{3328} a^{6} - \frac{165}{1664} a^{5} + \frac{81}{416} a^{4} - \frac{33}{208} a^{3} + \frac{61}{208} a^{2} + \frac{5}{52} a - \frac{1}{13}$, $\frac{1}{186119737244226195269137408} a^{15} - \frac{17133398485038792422603}{186119737244226195269137408} a^{14} + \frac{223549827558541005960565}{93059868622113097634568704} a^{13} - \frac{5420964980936400718314047}{186119737244226195269137408} a^{12} + \frac{460417428789904956888757}{14316902864940476559164416} a^{11} + \frac{10082989813173084734066509}{186119737244226195269137408} a^{10} + \frac{6005082021195893625907973}{46529934311056548817284352} a^{9} + \frac{60160797941602644322042449}{186119737244226195269137408} a^{8} - \frac{15522324819594949714450723}{186119737244226195269137408} a^{7} + \frac{15071782597428125622097125}{93059868622113097634568704} a^{6} - \frac{278562004097714572176701}{1454060447220517150540136} a^{5} + \frac{524599677162774552088341}{11632483577764137204321088} a^{4} - \frac{356538868089377818234607}{894806429058779784947776} a^{3} - \frac{5114440530135859208486}{13981350454043434139809} a^{2} - \frac{226199190844661666259197}{727030223610258575270068} a - \frac{74846935495119172162308}{181757555902564643817517}$

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

Class group and class number

$C_{140}$, which has order $140$ (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:  \( 3640.01221338 \) (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

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.0.437257.1, 4.0.25721.1, 8.0.3250292628833.2, \(\Q(\zeta_{17})^+\), 8.0.191193684049.1

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 ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$