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Properties

Label 16.0.23563340466...2849.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{10}\cdot 43^{8}$
Root discriminant $38.53$
Ramified primes $17, 43$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8957, -9958, 5911, -6308, 3602, -540, -209, 162, -59, 14, 187, -168, 94, -44, 11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 11*x^14 - 44*x^13 + 94*x^12 - 168*x^11 + 187*x^10 + 14*x^9 - 59*x^8 + 162*x^7 - 209*x^6 - 540*x^5 + 3602*x^4 - 6308*x^3 + 5911*x^2 - 9958*x + 8957)
 
gp: K = bnfinit(x^16 - 2*x^15 + 11*x^14 - 44*x^13 + 94*x^12 - 168*x^11 + 187*x^10 + 14*x^9 - 59*x^8 + 162*x^7 - 209*x^6 - 540*x^5 + 3602*x^4 - 6308*x^3 + 5911*x^2 - 9958*x + 8957, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 11 x^{14} - 44 x^{13} + 94 x^{12} - 168 x^{11} + 187 x^{10} + 14 x^{9} - 59 x^{8} + 162 x^{7} - 209 x^{6} - 540 x^{5} + 3602 x^{4} - 6308 x^{3} + 5911 x^{2} - 9958 x + 8957 \)

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:  \(23563340466869924558542849=17^{10}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{26} a^{10} + \frac{2}{13} a^{9} + \frac{1}{13} a^{8} - \frac{1}{13} a^{7} - \frac{4}{13} a^{6} - \frac{3}{26} a^{5} + \frac{4}{13} a^{4} - \frac{4}{13} a^{3} + \frac{2}{13} a^{2} + \frac{1}{13} a - \frac{1}{2}$, $\frac{1}{26} a^{11} - \frac{1}{26} a^{9} + \frac{3}{26} a^{8} - \frac{5}{13} a^{6} + \frac{7}{26} a^{5} - \frac{1}{26} a^{4} + \frac{5}{13} a^{3} - \frac{1}{26} a^{2} - \frac{4}{13} a$, $\frac{1}{26} a^{12} - \frac{3}{13} a^{9} + \frac{1}{13} a^{8} + \frac{1}{26} a^{7} + \frac{6}{13} a^{6} - \frac{2}{13} a^{5} - \frac{4}{13} a^{4} + \frac{2}{13} a^{3} + \frac{9}{26} a^{2} + \frac{1}{13} a$, $\frac{1}{338} a^{13} + \frac{1}{169} a^{12} + \frac{5}{338} a^{11} - \frac{1}{338} a^{10} + \frac{9}{169} a^{9} - \frac{24}{169} a^{8} + \frac{95}{338} a^{7} + \frac{47}{338} a^{6} + \frac{2}{169} a^{5} + \frac{75}{338} a^{4} - \frac{84}{169} a^{3} - \frac{2}{169} a^{2} - \frac{6}{13} a$, $\frac{1}{4394} a^{14} + \frac{5}{4394} a^{13} + \frac{38}{2197} a^{12} - \frac{19}{2197} a^{11} + \frac{40}{2197} a^{10} + \frac{471}{2197} a^{9} + \frac{450}{2197} a^{8} - \frac{627}{2197} a^{7} - \frac{1779}{4394} a^{6} - \frac{225}{4394} a^{5} + \frac{308}{2197} a^{4} - \frac{2133}{4394} a^{3} - \frac{623}{4394} a^{2} + \frac{29}{338} a - \frac{5}{13}$, $\frac{1}{14161815979244056526} a^{15} + \frac{21121952952559}{7080907989622028263} a^{14} + \frac{613728338906555}{1089370459941850502} a^{13} + \frac{104275064303240891}{14161815979244056526} a^{12} - \frac{109242582860330339}{7080907989622028263} a^{11} - \frac{135794702363224002}{7080907989622028263} a^{10} + \frac{3264369645936713127}{14161815979244056526} a^{9} - \frac{1570229260983376445}{14161815979244056526} a^{8} + \frac{3189494830395834212}{7080907989622028263} a^{7} + \frac{6615105696466916261}{14161815979244056526} a^{6} + \frac{1707215650273449865}{7080907989622028263} a^{5} - \frac{1474373258638717209}{7080907989622028263} a^{4} - \frac{1605117535078417686}{7080907989622028263} a^{3} - \frac{1459661218797321558}{7080907989622028263} a^{2} - \frac{106269500696947057}{544685229970925251} a + \frac{17497279125983928}{41898863843917327}$

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:  \( 964651.296901 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{-731}) \), 4.2.12427.1 x2, 4.0.31433.1 x2, \(\Q(\sqrt{17}, \sqrt{-43})\), 8.0.16796569313.1, 8.0.4854208531457.1, 8.0.285541678321.1

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 8 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$43$43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$