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Properties

Label 16.0.14319168638...3073.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 29^{8}$
Root discriminant $76.69$
Ramified primes $17, 29$
Class number $40834$ (GRH)
Class group $[40834]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![371265721, -273264104, 273264104, -105261332, 105261332, -21259946, 21259946, -2402492, 2402492, -157557, 157557, -5951, 5951, -120, 120, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 120*x^14 - 120*x^13 + 5951*x^12 - 5951*x^11 + 157557*x^10 - 157557*x^9 + 2402492*x^8 - 2402492*x^7 + 21259946*x^6 - 21259946*x^5 + 105261332*x^4 - 105261332*x^3 + 273264104*x^2 - 273264104*x + 371265721)
 
gp: K = bnfinit(x^16 - x^15 + 120*x^14 - 120*x^13 + 5951*x^12 - 5951*x^11 + 157557*x^10 - 157557*x^9 + 2402492*x^8 - 2402492*x^7 + 21259946*x^6 - 21259946*x^5 + 105261332*x^4 - 105261332*x^3 + 273264104*x^2 - 273264104*x + 371265721, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 120 x^{14} - 120 x^{13} + 5951 x^{12} - 5951 x^{11} + 157557 x^{10} - 157557 x^{9} + 2402492 x^{8} - 2402492 x^{7} + 21259946 x^{6} - 21259946 x^{5} + 105261332 x^{4} - 105261332 x^{3} + 273264104 x^{2} - 273264104 x + 371265721 \)

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:  \(1431916863894665085730117693073=17^{15}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.69$
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:  $17, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(493=17\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{493}(1,·)$, $\chi_{493}(260,·)$, $\chi_{493}(405,·)$, $\chi_{493}(407,·)$, $\chi_{493}(347,·)$, $\chi_{493}(28,·)$, $\chi_{493}(349,·)$, $\chi_{493}(30,·)$, $\chi_{493}(291,·)$, $\chi_{493}(231,·)$, $\chi_{493}(173,·)$, $\chi_{493}(117,·)$, $\chi_{493}(57,·)$, $\chi_{493}(378,·)$, $\chi_{493}(59,·)$, $\chi_{493}(318,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{69174631} a^{9} - \frac{2687102}{69174631} a^{8} + \frac{63}{69174631} a^{7} - \frac{12128450}{69174631} a^{6} + \frac{1323}{69174631} a^{5} - \frac{4723982}{69174631} a^{4} + \frac{10290}{69174631} a^{3} - \frac{12619373}{69174631} a^{2} + \frac{21609}{69174631} a + \frac{32192193}{69174631}$, $\frac{1}{69174631} a^{10} + \frac{70}{69174631} a^{8} + \frac{18809714}{69174631} a^{7} + \frac{1715}{69174631} a^{6} + \frac{22405783}{69174631} a^{5} + \frac{17150}{69174631} a^{4} - \frac{32192193}{69174631} a^{3} + \frac{60025}{69174631} a^{2} - \frac{8910729}{69174631} a + \frac{33614}{69174631}$, $\frac{1}{69174631} a^{11} - \frac{617039}{69174631} a^{8} - \frac{2695}{69174631} a^{7} - \frac{27872920}{69174631} a^{6} - \frac{75460}{69174631} a^{5} + \frac{21788023}{69174631} a^{4} - \frac{660275}{69174631} a^{3} - \frac{24824822}{69174631} a^{2} - \frac{1479016}{69174631} a + \frac{29309313}{69174631}$, $\frac{1}{69174631} a^{12} - \frac{3234}{69174631} a^{8} + \frac{11000537}{69174631} a^{7} - \frac{105644}{69174631} a^{6} + \frac{8035048}{69174631} a^{5} - \frac{1188495}{69174631} a^{4} + \frac{29615067}{69174631} a^{3} - \frac{4437048}{69174631} a^{2} + \frac{12201281}{69174631} a - \frac{2588278}{69174631}$, $\frac{1}{69174631} a^{13} - \frac{32258456}{69174631} a^{8} + \frac{98098}{69174631} a^{7} + \frac{6643525}{69174631} a^{6} + \frac{3090087}{69174631} a^{5} - \frac{29323901}{69174631} a^{4} + \frac{28840812}{69174631} a^{3} + \frac{14181289}{69174631} a^{2} - \frac{1879403}{69174631} a + \frac{1732507}{69174631}$, $\frac{1}{69174631} a^{14} + \frac{124852}{69174631} a^{8} + \frac{32861954}{69174631} a^{7} + \frac{4588311}{69174631} a^{6} - \frac{32133940}{69174631} a^{5} - \frac{14114899}{69174631} a^{4} - \frac{15360640}{69174631} a^{3} + \frac{6597287}{69174631} a^{2} + \frac{1951624}{69174631} a - \frac{9876554}{69174631}$, $\frac{1}{69174631} a^{15} + \frac{25960508}{69174631} a^{8} - \frac{3277365}{69174631} a^{7} - \frac{3567130}{69174631} a^{6} + \frac{28229798}{69174631} a^{5} + \frac{336118}{69174631} a^{4} - \frac{32986435}{69174631} a^{3} + \frac{34513764}{69174631} a^{2} - \frac{9992813}{69174631} a - \frac{6095443}{69174631}$

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

Class group and class number

$C_{40834}$, which has order $40834$ (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

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ R $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
29Data not computed