↑

Properties

Label 16.16.7336077203...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $3^{8}\cdot 5^{8}\cdot 17^{15}$
Root discriminant $55.16$
Ramified primes $3, 5, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_{16}$ (as 16T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-239699, 1353811, -1353811, -1988525, 1988525, 936019, -936019, -212909, 212909, 26451, -26451, -1837, 1837, 67, -67, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 67*x^14 + 67*x^13 + 1837*x^12 - 1837*x^11 - 26451*x^10 + 26451*x^9 + 212909*x^8 - 212909*x^7 - 936019*x^6 + 936019*x^5 + 1988525*x^4 - 1988525*x^3 - 1353811*x^2 + 1353811*x - 239699)
 
gp: K = bnfinit(x^16 - x^15 - 67*x^14 + 67*x^13 + 1837*x^12 - 1837*x^11 - 26451*x^10 + 26451*x^9 + 212909*x^8 - 212909*x^7 - 936019*x^6 + 936019*x^5 + 1988525*x^4 - 1988525*x^3 - 1353811*x^2 + 1353811*x - 239699, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 67 x^{14} + 67 x^{13} + 1837 x^{12} - 1837 x^{11} - 26451 x^{10} + 26451 x^{9} + 212909 x^{8} - 212909 x^{7} - 936019 x^{6} + 936019 x^{5} + 1988525 x^{4} - 1988525 x^{3} - 1353811 x^{2} + 1353811 x - 239699 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7336077203498398991356640625=3^{8}\cdot 5^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.16$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(194,·)$, $\chi_{255}(196,·)$, $\chi_{255}(74,·)$, $\chi_{255}(76,·)$, $\chi_{255}(14,·)$, $\chi_{255}(16,·)$, $\chi_{255}(209,·)$, $\chi_{255}(151,·)$, $\chi_{255}(29,·)$, $\chi_{255}(224,·)$, $\chi_{255}(164,·)$, $\chi_{255}(166,·)$, $\chi_{255}(106,·)$, $\chi_{255}(44,·)$, $\chi_{255}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{27403} a^{9} + \frac{3683}{27403} a^{8} - \frac{36}{27403} a^{7} - \frac{8244}{27403} a^{6} + \frac{432}{27403} a^{5} + \frac{231}{27403} a^{4} - \frac{1920}{27403} a^{3} + \frac{10222}{27403} a^{2} + \frac{2304}{27403} a - \frac{5111}{27403}$, $\frac{1}{27403} a^{10} - \frac{40}{27403} a^{8} - \frac{12671}{27403} a^{7} + \frac{560}{27403} a^{6} - \frac{1451}{27403} a^{5} - \frac{3200}{27403} a^{4} + \frac{11608}{27403} a^{3} + \frac{6400}{27403} a^{2} + \frac{4187}{27403} a - \frac{2048}{27403}$, $\frac{1}{27403} a^{11} - \frac{2366}{27403} a^{8} - \frac{880}{27403} a^{7} - \frac{2375}{27403} a^{6} - \frac{13323}{27403} a^{5} - \frac{6555}{27403} a^{4} + \frac{11809}{27403} a^{3} + \frac{2022}{27403} a^{2} + \frac{7903}{27403} a - \frac{12619}{27403}$, $\frac{1}{27403} a^{12} - \frac{1056}{27403} a^{8} - \frac{5342}{27403} a^{7} - \frac{7691}{27403} a^{6} + \frac{1646}{27403} a^{5} + \frac{10295}{27403} a^{4} + \frac{8200}{27403} a^{3} - \frac{3694}{27403} a^{2} + \frac{12851}{27403} a - \frac{7903}{27403}$, $\frac{1}{27403} a^{13} - \frac{7320}{27403} a^{8} + \frac{9099}{27403} a^{7} + \frac{10136}{27403} a^{6} + \frac{636}{27403} a^{5} + \frac{5509}{27403} a^{4} - \frac{3392}{27403} a^{3} + \frac{10501}{27403} a^{2} + \frac{13657}{27403} a + \frac{1175}{27403}$, $\frac{1}{27403} a^{14} + \frac{4107}{27403} a^{8} - \frac{6757}{27403} a^{7} - \frac{4038}{27403} a^{6} - \frac{10999}{27403} a^{5} - \frac{11458}{27403} a^{4} - \frac{13563}{27403} a^{3} + \frac{1104}{27403} a^{2} + \frac{13610}{27403} a - \frac{7425}{27403}$, $\frac{1}{27403} a^{15} - \frac{6382}{27403} a^{8} + \frac{6799}{27403} a^{7} + \frac{4404}{27403} a^{6} - \frac{4487}{27403} a^{5} - \frac{3175}{27403} a^{4} - \frac{5520}{27403} a^{3} + \frac{13252}{27403} a^{2} + \frac{11485}{27403} a + \frac{179}{27403}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 240645181.88595447 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3Data not computed
5Data not computed
17Data not computed