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Properties

Label 16.0.28596554320...5089.6
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 19^{8}$
Root discriminant $52.00$
Ramified primes $17, 19$
Class number $2600$ (GRH)
Class group $[5, 520]$ (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![3329789, -1635129, 2829154, -1299079, 1197274, -498338, 322825, -119410, 60042, -19323, 7860, -2142, 705, -152, 40, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 40*x^14 - 152*x^13 + 705*x^12 - 2142*x^11 + 7860*x^10 - 19323*x^9 + 60042*x^8 - 119410*x^7 + 322825*x^6 - 498338*x^5 + 1197274*x^4 - 1299079*x^3 + 2829154*x^2 - 1635129*x + 3329789)
 
gp: K = bnfinit(x^16 - 6*x^15 + 40*x^14 - 152*x^13 + 705*x^12 - 2142*x^11 + 7860*x^10 - 19323*x^9 + 60042*x^8 - 119410*x^7 + 322825*x^6 - 498338*x^5 + 1197274*x^4 - 1299079*x^3 + 2829154*x^2 - 1635129*x + 3329789, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 40 x^{14} - 152 x^{13} + 705 x^{12} - 2142 x^{11} + 7860 x^{10} - 19323 x^{9} + 60042 x^{8} - 119410 x^{7} + 322825 x^{6} - 498338 x^{5} + 1197274 x^{4} - 1299079 x^{3} + 2829154 x^{2} - 1635129 x + 3329789 \)

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:  \(2859655432078149808865465089=17^{14}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.00$
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:  $17, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(323=17\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{323}(1,·)$, $\chi_{323}(322,·)$, $\chi_{323}(132,·)$, $\chi_{323}(134,·)$, $\chi_{323}(77,·)$, $\chi_{323}(208,·)$, $\chi_{323}(18,·)$, $\chi_{323}(151,·)$, $\chi_{323}(94,·)$, $\chi_{323}(229,·)$, $\chi_{323}(172,·)$, $\chi_{323}(305,·)$, $\chi_{323}(115,·)$, $\chi_{323}(246,·)$, $\chi_{323}(189,·)$, $\chi_{323}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2626} a^{14} + \frac{14}{101} a^{13} + \frac{227}{2626} a^{12} - \frac{441}{2626} a^{11} + \frac{12}{101} a^{10} + \frac{957}{2626} a^{9} - \frac{87}{2626} a^{8} - \frac{328}{1313} a^{7} + \frac{577}{2626} a^{6} - \frac{83}{2626} a^{5} - \frac{453}{1313} a^{4} + \frac{1303}{2626} a^{3} - \frac{937}{2626} a^{2} - \frac{81}{1313} a - \frac{389}{2626}$, $\frac{1}{11345321689924520647363319372085902} a^{15} + \frac{285844173486812169806227218852}{5672660844962260323681659686042951} a^{14} + \frac{2420197997392174927215380534630961}{11345321689924520647363319372085902} a^{13} + \frac{2478625408084655032564336104390199}{11345321689924520647363319372085902} a^{12} + \frac{1243343185253452754263613163861724}{5672660844962260323681659686042951} a^{11} - \frac{4727466501356135661851083523540301}{11345321689924520647363319372085902} a^{10} - \frac{3074301274879329509847836259832149}{11345321689924520647363319372085902} a^{9} - \frac{2341290193967720233244012795584612}{5672660844962260323681659686042951} a^{8} + \frac{4520085226394908847807814009546847}{11345321689924520647363319372085902} a^{7} + \frac{4631101981051867996762071347469731}{11345321689924520647363319372085902} a^{6} + \frac{2350096542599862699671699802637765}{5672660844962260323681659686042951} a^{5} + \frac{83828282140663911560209504471581}{872717053071116972874101490160454} a^{4} + \frac{422829664333857838302916067769969}{872717053071116972874101490160454} a^{3} + \frac{1296784378327281706686082044853482}{5672660844962260323681659686042951} a^{2} - \frac{3580660173079134256203961755068495}{11345321689924520647363319372085902} a + \frac{1494293629905034420966102384668426}{5672660844962260323681659686042951}$

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

Class group and class number

$C_{5}\times C_{520}$, which has order $2600$ (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

$C_2\times C_8$ (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

\(\Q(\sqrt{-323}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{17}, \sqrt{-19})\), 4.4.4913.1, 4.0.1773593.2, 8.0.3145632129649.3, 8.0.53475746204033.2, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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}$
$19$19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$