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Properties

Label 16.0.37916588395...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 7^{8}\cdot 17^{14}$
Root discriminant $70.58$
Ramified primes $5, 7, 17$
Class number $24272$ (GRH)
Class group $[24272]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![160839449, -61631729, 98545822, -33852655, 28370042, -8598346, 4968321, -1307274, 576030, -128635, 45140, -8230, 2329, -320, 72, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 72*x^14 - 320*x^13 + 2329*x^12 - 8230*x^11 + 45140*x^10 - 128635*x^9 + 576030*x^8 - 1307274*x^7 + 4968321*x^6 - 8598346*x^5 + 28370042*x^4 - 33852655*x^3 + 98545822*x^2 - 61631729*x + 160839449)
 
gp: K = bnfinit(x^16 - 6*x^15 + 72*x^14 - 320*x^13 + 2329*x^12 - 8230*x^11 + 45140*x^10 - 128635*x^9 + 576030*x^8 - 1307274*x^7 + 4968321*x^6 - 8598346*x^5 + 28370042*x^4 - 33852655*x^3 + 98545822*x^2 - 61631729*x + 160839449, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 72 x^{14} - 320 x^{13} + 2329 x^{12} - 8230 x^{11} + 45140 x^{10} - 128635 x^{9} + 576030 x^{8} - 1307274 x^{7} + 4968321 x^{6} - 8598346 x^{5} + 28370042 x^{4} - 33852655 x^{3} + 98545822 x^{2} - 61631729 x + 160839449 \)

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:  \(379165883956039466757862890625=5^{8}\cdot 7^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.58$
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:  $5, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(595=5\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{595}(1,·)$, $\chi_{595}(69,·)$, $\chi_{595}(526,·)$, $\chi_{595}(594,·)$, $\chi_{595}(281,·)$, $\chi_{595}(349,·)$, $\chi_{595}(36,·)$, $\chi_{595}(421,·)$, $\chi_{595}(104,·)$, $\chi_{595}(489,·)$, $\chi_{595}(106,·)$, $\chi_{595}(491,·)$, $\chi_{595}(174,·)$, $\chi_{595}(559,·)$, $\chi_{595}(246,·)$, $\chi_{595}(314,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{26} a^{14} + \frac{3}{26} a^{12} - \frac{3}{26} a^{11} - \frac{1}{13} a^{10} - \frac{1}{26} a^{9} - \frac{7}{26} a^{8} - \frac{6}{13} a^{7} - \frac{5}{26} a^{6} + \frac{11}{26} a^{5} + \frac{1}{13} a^{4} - \frac{9}{26} a^{3} - \frac{11}{26} a^{2} + \frac{2}{13} a + \frac{1}{26}$, $\frac{1}{3501681333523774670297315096503237679325502} a^{15} - \frac{12388771777331603902650875737688293025943}{3501681333523774670297315096503237679325502} a^{14} + \frac{13254026112621838371920559344659326218864}{1750840666761887335148657548251618839662751} a^{13} - \frac{288224172903697899243705503142632480373724}{1750840666761887335148657548251618839662751} a^{12} + \frac{992178897181916211711925853549706325365711}{3501681333523774670297315096503237679325502} a^{11} - \frac{55426081540927355623037877657548169355271}{134680051289375948857589042173201449204827} a^{10} - \frac{27732446326920986242657937873021076302235}{134680051289375948857589042173201449204827} a^{9} + \frac{1718118118166115728190369308083155562768551}{3501681333523774670297315096503237679325502} a^{8} + \frac{222268571840664304338014296021271640997460}{1750840666761887335148657548251618839662751} a^{7} - \frac{577606995713662053166119078941255202037204}{1750840666761887335148657548251618839662751} a^{6} - \frac{191970533416614872184485879116068980503833}{3501681333523774670297315096503237679325502} a^{5} + \frac{472094515039173172717791171900249380080802}{1750840666761887335148657548251618839662751} a^{4} + \frac{49734471539457239531923653677946084623500}{134680051289375948857589042173201449204827} a^{3} + \frac{896456488416506451737402017647229822879301}{3501681333523774670297315096503237679325502} a^{2} + \frac{106380323095169321626007292288771731823657}{1750840666761887335148657548251618839662751} a - \frac{891381654668733005273492768476309585997231}{3501681333523774670297315096503237679325502}$

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

Class group and class number

$C_{24272}$, which has order $24272$ (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:  \( 3640.012213375973 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-595}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{17}, \sqrt{-35})\), 4.4.4913.1, 4.0.6018425.1, 8.0.36221439480625.7, \(\Q(\zeta_{17})^+\), 8.0.615764471170625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ 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
5Data not computed
7Data not computed
$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}$