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Properties

Label 16.0.54874743186...8833.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 61^{8}$
Root discriminant $111.23$
Ramified primes $17, 61$
Class number $1265378$ (GRH)
Class group $[1265378]$ (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![87458231281, -43889090656, 43889090656, -9033778156, 9033778156, -900871906, 900871906, -48853156, 48853156, -1518781, 1518781, -27031, 27031, -256, 256, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 256*x^14 - 256*x^13 + 27031*x^12 - 27031*x^11 + 1518781*x^10 - 1518781*x^9 + 48853156*x^8 - 48853156*x^7 + 900871906*x^6 - 900871906*x^5 + 9033778156*x^4 - 9033778156*x^3 + 43889090656*x^2 - 43889090656*x + 87458231281)
 
gp: K = bnfinit(x^16 - x^15 + 256*x^14 - 256*x^13 + 27031*x^12 - 27031*x^11 + 1518781*x^10 - 1518781*x^9 + 48853156*x^8 - 48853156*x^7 + 900871906*x^6 - 900871906*x^5 + 9033778156*x^4 - 9033778156*x^3 + 43889090656*x^2 - 43889090656*x + 87458231281, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 256 x^{14} - 256 x^{13} + 27031 x^{12} - 27031 x^{11} + 1518781 x^{10} - 1518781 x^{9} + 48853156 x^{8} - 48853156 x^{7} + 900871906 x^{6} - 900871906 x^{5} + 9033778156 x^{4} - 9033778156 x^{3} + 43889090656 x^{2} - 43889090656 x + 87458231281 \)

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:  \(548747431866424450523939019858833=17^{15}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.23$
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:  $17, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1037=17\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{1037}(1,·)$, $\chi_{1037}(975,·)$, $\chi_{1037}(977,·)$, $\chi_{1037}(916,·)$, $\chi_{1037}(853,·)$, $\chi_{1037}(792,·)$, $\chi_{1037}(733,·)$, $\chi_{1037}(670,·)$, $\chi_{1037}(672,·)$, $\chi_{1037}(609,·)$, $\chi_{1037}(611,·)$, $\chi_{1037}(487,·)$, $\chi_{1037}(489,·)$, $\chi_{1037}(243,·)$, $\chi_{1037}(182,·)$, $\chi_{1037}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11482735951} a^{9} + \frac{4393463955}{11482735951} a^{8} + \frac{135}{11482735951} a^{7} - \frac{990179146}{11482735951} a^{6} + \frac{6075}{11482735951} a^{5} - \frac{2683510122}{11482735951} a^{4} + \frac{101250}{11482735951} a^{3} + \frac{2246086389}{11482735951} a^{2} + \frac{455625}{11482735951} a - \frac{2965297990}{11482735951}$, $\frac{1}{11482735951} a^{10} + \frac{150}{11482735951} a^{8} + \frac{2994456381}{11482735951} a^{7} + \frac{7875}{11482735951} a^{6} + \frac{4384049328}{11482735951} a^{5} + \frac{168750}{11482735951} a^{4} + \frac{5211384379}{11482735951} a^{3} + \frac{1265625}{11482735951} a^{2} - \frac{1104192986}{11482735951} a + \frac{1518750}{11482735951}$, $\frac{1}{11482735951} a^{11} - \frac{1509187662}{11482735951} a^{8} - \frac{12375}{11482735951} a^{7} + \frac{3635353865}{11482735951} a^{6} - \frac{742500}{11482735951} a^{5} - \frac{5640591557}{11482735951} a^{4} - \frac{13921875}{11482735951} a^{3} - \frac{5017808757}{11482735951} a^{2} - \frac{66825000}{11482735951} a - \frac{3032003589}{11482735951}$, $\frac{1}{11482735951} a^{12} - \frac{14850}{11482735951} a^{8} + \frac{686441117}{11482735951} a^{7} - \frac{1039500}{11482735951} a^{6} - \frac{548833805}{11482735951} a^{5} - \frac{25059375}{11482735951} a^{4} - \frac{534331214}{11482735951} a^{3} - \frac{200475000}{11482735951} a^{2} - \frac{80458572}{11482735951} a - \frac{250593750}{11482735951}$, $\frac{1}{11482735951} a^{13} - \frac{1279500715}{11482735951} a^{8} + \frac{965250}{11482735951} a^{7} + \frac{4675601326}{11482735951} a^{6} + \frac{65154375}{11482735951} a^{5} - \frac{5565892944}{11482735951} a^{4} + \frac{1303087500}{11482735951} a^{3} - \frac{3045519577}{11482735951} a^{2} - \frac{4967298451}{11482735951} a + \frac{1617220585}{11482735951}$, $\frac{1}{11482735951} a^{14} + \frac{1228500}{11482735951} a^{8} + \frac{5167158586}{11482735951} a^{7} + \frac{96744375}{11482735951} a^{6} + \frac{5071447805}{11482735951} a^{5} + \frac{2487712500}{11482735951} a^{4} - \frac{1825125009}{11482735951} a^{3} - \frac{2234534402}{11482735951} a^{2} - \frac{4373739810}{11482735951} a + \frac{3688590598}{11482735951}$, $\frac{1}{11482735951} a^{15} + \frac{2870321028}{11482735951} a^{8} - \frac{69103125}{11482735951} a^{7} + \frac{5036603669}{11482735951} a^{6} - \frac{4975425000}{11482735951} a^{5} - \frac{3131780109}{11482735951} a^{4} - \frac{310063941}{11482735951} a^{3} + \frac{2911870892}{11482735951} a^{2} - \frac{4875396254}{11482735951} a + \frac{5048468103}{11482735951}$

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

Class group and class number

$C_{1265378}$, which has order $1265378$ (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 ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
61Data not computed