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Properties

Label 16.0.10054202156...4353.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 37^{8}$
Root discriminant $86.63$
Ramified primes $17, 37$
Class number $83234$ (GRH)
Class group $[83234]$ (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![2159688331, -1427894074, 1427894074, -452168398, 452168398, -72719524, 72719524, -6466546, 6466546, -332011, 332011, -9793, 9793, -154, 154, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 154*x^14 - 154*x^13 + 9793*x^12 - 9793*x^11 + 332011*x^10 - 332011*x^9 + 6466546*x^8 - 6466546*x^7 + 72719524*x^6 - 72719524*x^5 + 452168398*x^4 - 452168398*x^3 + 1427894074*x^2 - 1427894074*x + 2159688331)
 
gp: K = bnfinit(x^16 - x^15 + 154*x^14 - 154*x^13 + 9793*x^12 - 9793*x^11 + 332011*x^10 - 332011*x^9 + 6466546*x^8 - 6466546*x^7 + 72719524*x^6 - 72719524*x^5 + 452168398*x^4 - 452168398*x^3 + 1427894074*x^2 - 1427894074*x + 2159688331, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 154 x^{14} - 154 x^{13} + 9793 x^{12} - 9793 x^{11} + 332011 x^{10} - 332011 x^{9} + 6466546 x^{8} - 6466546 x^{7} + 72719524 x^{6} - 72719524 x^{5} + 452168398 x^{4} - 452168398 x^{3} + 1427894074 x^{2} - 1427894074 x + 2159688331 \)

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:  \(10054202156858080231167941574353=17^{15}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.63$
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:  $17, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(629=17\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{629}(1,·)$, $\chi_{629}(258,·)$, $\chi_{629}(517,·)$, $\chi_{629}(519,·)$, $\chi_{629}(73,·)$, $\chi_{629}(593,·)$, $\chi_{629}(147,·)$, $\chi_{629}(149,·)$, $\chi_{629}(223,·)$, $\chi_{629}(38,·)$, $\chi_{629}(295,·)$, $\chi_{629}(297,·)$, $\chi_{629}(554,·)$, $\chi_{629}(369,·)$, $\chi_{629}(184,·)$, $\chi_{629}(186,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{357580549} a^{9} - \frac{59581000}{357580549} a^{8} + \frac{81}{357580549} a^{7} + \frac{1134588}{357580549} a^{6} + \frac{2187}{357580549} a^{5} + \frac{25528230}{357580549} a^{4} + \frac{21870}{357580549} a^{3} - \frac{173777293}{357580549} a^{2} + \frac{59049}{357580549} a - \frac{150801886}{357580549}$, $\frac{1}{357580549} a^{10} + \frac{90}{357580549} a^{8} + \frac{178648451}{357580549} a^{7} + \frac{2835}{357580549} a^{6} + \frac{169855394}{357580549} a^{5} + \frac{36450}{357580549} a^{4} - \frac{160827849}{357580549} a^{3} + \frac{164025}{357580549} a^{2} + \frac{170226052}{357580549} a + \frac{118098}{357580549}$, $\frac{1}{357580549} a^{11} + \frac{177230216}{357580549} a^{8} - \frac{4455}{357580549} a^{7} + \frac{67742474}{357580549} a^{6} - \frac{160380}{357580549} a^{5} + \frac{44695294}{357580549} a^{4} - \frac{1804275}{357580549} a^{3} + \frac{76638266}{357580549} a^{2} - \frac{5196312}{357580549} a - \frac{15891122}{357580549}$, $\frac{1}{357580549} a^{12} - \frac{5346}{357580549} a^{8} + \frac{15316938}{357580549} a^{7} - \frac{224532}{357580549} a^{6} + \frac{59528018}{357580549} a^{5} - \frac{3247695}{357580549} a^{4} - \frac{132615043}{357580549} a^{3} - \frac{15588936}{357580549} a^{2} + \frac{27011877}{357580549} a - \frac{11691702}{357580549}$, $\frac{1}{357580549} a^{13} + \frac{99560097}{357580549} a^{8} + \frac{208494}{357580549} a^{7} + \frac{46166133}{357580549} a^{6} + \frac{8444007}{357580549} a^{5} + \frac{103113368}{357580549} a^{4} + \frac{101328084}{357580549} a^{3} + \frac{7869801}{357580549} a^{2} - \frac{53596297}{357580549} a + \frac{157255439}{357580549}$, $\frac{1}{357580549} a^{14} + \frac{265356}{357580549} a^{8} - \frac{151429646}{357580549} a^{7} + \frac{12538071}{357580549} a^{6} + \frac{131735570}{357580549} a^{5} - \frac{164136025}{357580549} a^{4} - \frac{63488728}{357580549} a^{3} - \frac{105519027}{357580549} a^{2} - \frac{142686754}{357580549} a + \frac{30982066}{357580549}$, $\frac{1}{357580549} a^{15} - \frac{41987132}{357580549} a^{8} - \frac{8955765}{357580549} a^{7} + \frac{144824500}{357580549} a^{6} - \frac{29308499}{357580549} a^{5} - \frac{126568352}{357580549} a^{4} + \frac{170014586}{357580549} a^{3} - \frac{167763388}{357580549} a^{2} + \frac{95519778}{357580549} a + \frac{61183924}{357580549}$

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

Class group and class number

$C_{83234}$, which has order $83234$ (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 ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ R $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
37Data not computed