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Properties

Label 16.0.23084299571...1233.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 73^{8}$
Root discriminant $121.68$
Ramified primes $17, 73$
Class number $1356226$ (GRH)
Class group $[1356226]$ (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![338737808719, -151398478927, 151398478927, -26505592399, 26505592399, -2220864463, 2220864463, -100769167, 100769167, -2616607, 2616607, -38863, 38863, -307, 307, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 307*x^14 - 307*x^13 + 38863*x^12 - 38863*x^11 + 2616607*x^10 - 2616607*x^9 + 100769167*x^8 - 100769167*x^7 + 2220864463*x^6 - 2220864463*x^5 + 26505592399*x^4 - 26505592399*x^3 + 151398478927*x^2 - 151398478927*x + 338737808719)
 
gp: K = bnfinit(x^16 - x^15 + 307*x^14 - 307*x^13 + 38863*x^12 - 38863*x^11 + 2616607*x^10 - 2616607*x^9 + 100769167*x^8 - 100769167*x^7 + 2220864463*x^6 - 2220864463*x^5 + 26505592399*x^4 - 26505592399*x^3 + 151398478927*x^2 - 151398478927*x + 338737808719, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 307 x^{14} - 307 x^{13} + 38863 x^{12} - 38863 x^{11} + 2616607 x^{10} - 2616607 x^{9} + 100769167 x^{8} - 100769167 x^{7} + 2220864463 x^{6} - 2220864463 x^{5} + 26505592399 x^{4} - 26505592399 x^{3} + 151398478927 x^{2} - 151398478927 x + 338737808719 \)

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:  \(2308429957160341796132964777021233=17^{15}\cdot 73^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.68$
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:  $17, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1241=17\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{1241}(512,·)$, $\chi_{1241}(1,·)$, $\chi_{1241}(1094,·)$, $\chi_{1241}(583,·)$, $\chi_{1241}(1096,·)$, $\chi_{1241}(1167,·)$, $\chi_{1241}(656,·)$, $\chi_{1241}(1169,·)$, $\chi_{1241}(218,·)$, $\chi_{1241}(220,·)$, $\chi_{1241}(802,·)$, $\chi_{1241}(293,·)$, $\chi_{1241}(364,·)$, $\chi_{1241}(366,·)$, $\chi_{1241}(437,·)$, $\chi_{1241}(950,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{41129090011} a^{9} - \frac{8474963750}{41129090011} a^{8} + \frac{162}{41129090011} a^{7} + \frac{13477920330}{41129090011} a^{6} + \frac{8748}{41129090011} a^{5} - \frac{10429935315}{41129090011} a^{4} + \frac{174960}{41129090011} a^{3} + \frac{14325291508}{41129090011} a^{2} + \frac{944784}{41129090011} a - \frac{8897184118}{41129090011}$, $\frac{1}{41129090011} a^{10} + \frac{180}{41129090011} a^{8} - \frac{11967012544}{41129090011} a^{7} + \frac{11340}{41129090011} a^{6} + \frac{13932749863}{41129090011} a^{5} + \frac{291600}{41129090011} a^{4} + \frac{8029914936}{41129090011} a^{3} + \frac{2624400}{41129090011} a^{2} - \frac{9988945598}{41129090011} a + \frac{3779136}{41129090011}$, $\frac{1}{41129090011} a^{11} - \frac{8249867951}{41129090011} a^{8} - \frac{17820}{41129090011} a^{7} + \frac{14523401112}{41129090011} a^{6} - \frac{1283040}{41129090011} a^{5} - \frac{6519868870}{41129090011} a^{4} - \frac{28868400}{41129090011} a^{3} + \frac{2591253655}{41129090011} a^{2} - \frac{166281984}{41129090011} a - \frac{2541369189}{41129090011}$, $\frac{1}{41129090011} a^{12} - \frac{21384}{41129090011} a^{8} - \frac{6257961189}{41129090011} a^{7} - \frac{1796256}{41129090011} a^{6} - \frac{18228002827}{41129090011} a^{5} - \frac{51963120}{41129090011} a^{4} + \frac{15203114581}{41129090011} a^{3} - \frac{498845952}{41129090011} a^{2} + \frac{7981953796}{41129090011} a - \frac{748268928}{41129090011}$, $\frac{1}{41129090011} a^{13} - \frac{20112202723}{41129090011} a^{8} + \frac{1667952}{41129090011} a^{7} + \frac{2086626816}{41129090011} a^{6} + \frac{135104112}{41129090011} a^{5} - \frac{16607621737}{41129090011} a^{4} + \frac{3242498688}{41129090011} a^{3} + \frac{10553158940}{41129090011} a^{2} + \frac{19454992128}{41129090011} a + \frac{5785211574}{41129090011}$, $\frac{1}{41129090011} a^{14} + \frac{2122848}{41129090011} a^{8} + \frac{11065357073}{41129090011} a^{7} + \frac{200609136}{41129090011} a^{6} + \frac{15823822020}{41129090011} a^{5} + \frac{6190224768}{41129090011} a^{4} + \frac{1116593904}{41129090011} a^{3} - \frac{20355932342}{41129090011} a^{2} + \frac{12408486395}{41129090011} a + \frac{13248144970}{41129090011}$, $\frac{1}{41129090011} a^{15} + \frac{14197695354}{41129090011} a^{8} - \frac{143292240}{41129090011} a^{7} + \frac{14460544363}{41129090011} a^{6} - \frac{12380449536}{41129090011} a^{5} - \frac{19101810650}{41129090011} a^{4} + \frac{19521481688}{41129090011} a^{3} - \frac{7284545110}{41129090011} a^{2} - \frac{18188359334}{41129090011} a - \frac{13462503378}{41129090011}$

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

Class group and class number

$C_{1356226}$, which has order $1356226$ (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
73Data not computed