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Properties

Label 16.16.6445820027...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{8}\cdot 7^{8}\cdot 17^{15}$
Root discriminant $84.25$
Ramified primes $5, 7, 17$
Class number $2$ (GRH)
Class group $[2]$ (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![75086281, 656707976, -656707976, -319017700, 319017700, 60431174, -60431174, -5821804, 5821804, 312731, -312731, -9487, 9487, 152, -152, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 152*x^14 + 152*x^13 + 9487*x^12 - 9487*x^11 - 312731*x^10 + 312731*x^9 + 5821804*x^8 - 5821804*x^7 - 60431174*x^6 + 60431174*x^5 + 319017700*x^4 - 319017700*x^3 - 656707976*x^2 + 656707976*x + 75086281)
 
gp: K = bnfinit(x^16 - x^15 - 152*x^14 + 152*x^13 + 9487*x^12 - 9487*x^11 - 312731*x^10 + 312731*x^9 + 5821804*x^8 - 5821804*x^7 - 60431174*x^6 + 60431174*x^5 + 319017700*x^4 - 319017700*x^3 - 656707976*x^2 + 656707976*x + 75086281, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 152 x^{14} + 152 x^{13} + 9487 x^{12} - 9487 x^{11} - 312731 x^{10} + 312731 x^{9} + 5821804 x^{8} - 5821804 x^{7} - 60431174 x^{6} + 60431174 x^{5} + 319017700 x^{4} - 319017700 x^{3} - 656707976 x^{2} + 656707976 x + 75086281 \)

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:  \(6445820027252670934883669140625=5^{8}\cdot 7^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.25$
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:  $5, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(595=5\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{595}(384,·)$, $\chi_{595}(1,·)$, $\chi_{595}(454,·)$, $\chi_{595}(139,·)$, $\chi_{595}(524,·)$, $\chi_{595}(526,·)$, $\chi_{595}(209,·)$, $\chi_{595}(279,·)$, $\chi_{595}(281,·)$, $\chi_{595}(419,·)$, $\chi_{595}(36,·)$, $\chi_{595}(421,·)$, $\chi_{595}(106,·)$, $\chi_{595}(491,·)$, $\chi_{595}(244,·)$, $\chi_{595}(246,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{41771753} a^{9} - \frac{8571045}{41771753} a^{8} - \frac{81}{41771753} a^{7} - \frac{9461055}{41771753} a^{6} + \frac{2187}{41771753} a^{5} - \frac{16870904}{41771753} a^{4} - \frac{21870}{41771753} a^{3} + \frac{12863951}{41771753} a^{2} + \frac{59049}{41771753} a - \frac{19693414}{41771753}$, $\frac{1}{41771753} a^{10} - \frac{90}{41771753} a^{8} + \frac{6404101}{41771753} a^{7} + \frac{2835}{41771753} a^{6} + \frac{14259167}{41771753} a^{5} - \frac{36450}{41771753} a^{4} - \frac{6034488}{41771753} a^{3} + \frac{164025}{41771753} a^{2} - \frac{14616557}{41771753} a - \frac{118098}{41771753}$, $\frac{1}{41771753} a^{11} - \frac{13098395}{41771753} a^{8} - \frac{4455}{41771753} a^{7} - \frac{1800723}{41771753} a^{6} + \frac{160380}{41771753} a^{5} - \frac{20632740}{41771753} a^{4} - \frac{1804275}{41771753} a^{3} + \frac{15301702}{41771753} a^{2} + \frac{5196312}{41771753} a - \frac{17993634}{41771753}$, $\frac{1}{41771753} a^{12} - \frac{5346}{41771753} a^{8} - \frac{18476893}{41771753} a^{7} + \frac{224532}{41771753} a^{6} + \frac{11906320}{41771753} a^{5} - \frac{3247695}{41771753} a^{4} - \frac{17686627}{41771753} a^{3} + \frac{15588936}{41771753} a^{2} - \frac{16645827}{41771753} a - \frac{11691702}{41771753}$, $\frac{1}{41771753} a^{13} - \frac{15670422}{41771753} a^{8} - \frac{208494}{41771753} a^{7} + \frac{18699173}{41771753} a^{6} + \frac{8444007}{41771753} a^{5} + \frac{17447069}{41771753} a^{4} - \frac{17784578}{41771753} a^{3} - \frac{2269219}{41771753} a^{2} + \frac{11581981}{41771753} a - \frac{16173684}{41771753}$, $\frac{1}{41771753} a^{14} - \frac{265356}{41771753} a^{8} + \frac{2547581}{41771753} a^{7} + \frac{12538071}{41771753} a^{6} - \frac{5949230}{41771753} a^{5} + \frac{15414241}{41771753} a^{4} - \frac{18936747}{41771753} a^{3} + \frac{6472301}{41771753} a^{2} + \frac{20474291}{41771753} a + \frac{5748390}{41771753}$, $\frac{1}{41771753} a^{15} + \frac{12737905}{41771753} a^{8} - \frac{8955765}{41771753} a^{7} + \frac{12238996}{41771753} a^{6} + \frac{10943271}{41771753} a^{5} - \frac{10454302}{41771753} a^{4} + \frac{9410248}{41771753} a^{3} + \frac{1172440}{41771753} a^{2} + \frac{10347459}{41771753} a + \frac{6050175}{41771753}$

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:  \( 7981880078.469981 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R R $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ 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
5Data not computed
7Data not computed
17Data not computed