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Properties

Label 16.0.17325691008...5616.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 17^{4}\cdot 16673^{4}$
Root discriminant $77.61$
Ramified primes $2, 17, 16673$
Class number $396$ (GRH)
Class group $[2, 198]$ (GRH)
Galois group 16T969

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15584, -25440, 29980, 56344, -14400, -10064, 14227, 4238, 515, -340, -384, 132, 52, -4, -7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 7*x^14 - 4*x^13 + 52*x^12 + 132*x^11 - 384*x^10 - 340*x^9 + 515*x^8 + 4238*x^7 + 14227*x^6 - 10064*x^5 - 14400*x^4 + 56344*x^3 + 29980*x^2 - 25440*x + 15584)
 
gp: K = bnfinit(x^16 - 2*x^15 - 7*x^14 - 4*x^13 + 52*x^12 + 132*x^11 - 384*x^10 - 340*x^9 + 515*x^8 + 4238*x^7 + 14227*x^6 - 10064*x^5 - 14400*x^4 + 56344*x^3 + 29980*x^2 - 25440*x + 15584, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 7 x^{14} - 4 x^{13} + 52 x^{12} + 132 x^{11} - 384 x^{10} - 340 x^{9} + 515 x^{8} + 4238 x^{7} + 14227 x^{6} - 10064 x^{5} - 14400 x^{4} + 56344 x^{3} + 29980 x^{2} - 25440 x + 15584 \)

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:  \(1732569100827059303563569135616=2^{28}\cdot 17^{4}\cdot 16673^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.61$
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:  $2, 17, 16673$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{68} a^{11} + \frac{3}{68} a^{10} + \frac{3}{68} a^{9} + \frac{3}{68} a^{8} - \frac{9}{68} a^{7} + \frac{9}{68} a^{6} - \frac{33}{68} a^{5} - \frac{33}{68} a^{4} - \frac{1}{2} a^{3} - \frac{15}{34} a^{2} + \frac{6}{17} a - \frac{6}{17}$, $\frac{1}{136} a^{12} - \frac{3}{68} a^{10} - \frac{3}{68} a^{9} + \frac{2}{17} a^{8} + \frac{1}{68} a^{7} + \frac{1}{17} a^{6} - \frac{1}{68} a^{5} + \frac{31}{136} a^{4} - \frac{15}{68} a^{3} + \frac{23}{68} a^{2} - \frac{7}{34} a - \frac{8}{17}$, $\frac{1}{272} a^{13} - \frac{1}{272} a^{12} - \frac{1}{136} a^{11} - \frac{11}{136} a^{10} - \frac{9}{68} a^{8} - \frac{4}{17} a^{7} + \frac{15}{68} a^{6} + \frac{71}{272} a^{5} - \frac{91}{272} a^{4} + \frac{21}{136} a^{3} + \frac{5}{136} a^{2} + \frac{8}{17} a - \frac{2}{17}$, $\frac{1}{4624} a^{14} + \frac{1}{2312} a^{13} - \frac{3}{4624} a^{12} - \frac{3}{578} a^{11} + \frac{273}{2312} a^{10} + \frac{25}{1156} a^{9} - \frac{253}{1156} a^{8} - \frac{41}{1156} a^{7} - \frac{717}{4624} a^{6} + \frac{1013}{2312} a^{5} + \frac{2079}{4624} a^{4} - \frac{151}{1156} a^{3} + \frac{1017}{2312} a^{2} - \frac{19}{578} a + \frac{99}{289}$, $\frac{1}{1879444981189011204139936} a^{15} - \frac{166648770307788555}{3251634915551922498512} a^{14} + \frac{2583190607796812099969}{1879444981189011204139936} a^{13} - \frac{2451420036472395330997}{939722490594505602069968} a^{12} + \frac{2535789302423829320769}{939722490594505602069968} a^{11} + \frac{8769102282102899622315}{234930622648626400517492} a^{10} - \frac{46716582287447053335383}{469861245297252801034984} a^{9} + \frac{42401035466565727732277}{469861245297252801034984} a^{8} + \frac{284702975102755444002827}{1879444981189011204139936} a^{7} - \frac{210277012984408959302635}{939722490594505602069968} a^{6} + \frac{681632756335850560985331}{1879444981189011204139936} a^{5} + \frac{200873441750189796486183}{939722490594505602069968} a^{4} - \frac{221555678624737340751803}{939722490594505602069968} a^{3} - \frac{208356765757492141071033}{469861245297252801034984} a^{2} + \frac{11156889313516568435121}{117465311324313200258746} a + \frac{16687466648497953057659}{58732655662156600129373}$

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

Class group and class number

$C_{2}\times C_{198}$, which has order $396$ (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:  \( 37998382.295 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T969:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 44 conjugacy class representatives for t16n969
Character table for t16n969 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.1067072.2, 8.0.77427700416512.1, 8.0.77427700416512.2, 8.8.329067726770176.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
16673Data not computed