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Properties

Label 16.16.9332481685...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $3^{8}\cdot 5^{12}\cdot 17^{12}$
Root discriminant $48.49$
Ramified primes $3, 5, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-89, -1571, -5992, 8755, 21775, -13913, -26965, 9019, 15456, -2665, -4475, 346, 648, -16, -43, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 43*x^14 - 16*x^13 + 648*x^12 + 346*x^11 - 4475*x^10 - 2665*x^9 + 15456*x^8 + 9019*x^7 - 26965*x^6 - 13913*x^5 + 21775*x^4 + 8755*x^3 - 5992*x^2 - 1571*x - 89)
 
gp: K = bnfinit(x^16 - 43*x^14 - 16*x^13 + 648*x^12 + 346*x^11 - 4475*x^10 - 2665*x^9 + 15456*x^8 + 9019*x^7 - 26965*x^6 - 13913*x^5 + 21775*x^4 + 8755*x^3 - 5992*x^2 - 1571*x - 89, 1)
 

Normalized defining polynomial

\( x^{16} - 43 x^{14} - 16 x^{13} + 648 x^{12} + 346 x^{11} - 4475 x^{10} - 2665 x^{9} + 15456 x^{8} + 9019 x^{7} - 26965 x^{6} - 13913 x^{5} + 21775 x^{4} + 8755 x^{3} - 5992 x^{2} - 1571 x - 89 \)

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:  \(933248168570425273681640625=3^{8}\cdot 5^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.49$
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$$\chi_{255}(64,·)$, $\chi_{255}(1,·)$, $\chi_{255}(4,·)$, $\chi_{255}(137,·)$, $\chi_{255}(203,·)$, $\chi_{255}(16,·)$, $\chi_{255}(152,·)$, $\chi_{255}(154,·)$, $\chi_{255}(98,·)$, $\chi_{255}(38,·)$, $\chi_{255}(166,·)$, $\chi_{255}(169,·)$, $\chi_{255}(106,·)$, $\chi_{255}(47,·)$, $\chi_{255}(242,·)$, $\chi_{255}(188,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{1181103214204970788724} a^{15} - \frac{35192449263720696605}{590551607102485394362} a^{14} - \frac{19612161626316419141}{295275803551242697181} a^{13} + \frac{51066608170495704379}{590551607102485394362} a^{12} + \frac{169830536150268968499}{1181103214204970788724} a^{11} + \frac{286239471445302601727}{1181103214204970788724} a^{10} + \frac{319071472206206935}{62163327063419515196} a^{9} + \frac{56800722690668685359}{1181103214204970788724} a^{8} - \frac{27005077882262107143}{1181103214204970788724} a^{7} + \frac{82617967549207623041}{1181103214204970788724} a^{6} + \frac{111040852465942480505}{295275803551242697181} a^{5} - \frac{184348244638815310873}{1181103214204970788724} a^{4} + \frac{253216407338279955419}{1181103214204970788724} a^{3} - \frac{89663615448016421349}{1181103214204970788724} a^{2} - \frac{10631615192340058347}{31081663531709757598} a - \frac{948040361391092649}{6635411315758262858}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 179468398.8534908 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), 4.4.5527125.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.5527125.1, 4.4.325125.1, \(\Q(\zeta_{15})^+\), 4.4.4913.1, 4.4.122825.1, 8.8.30549110765625.1, 8.8.105706265625.1, 8.8.15085980625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
17Data not computed