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Properties

Label 16.0.47991168245...0000.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 17^{8}$
Root discriminant $95.52$
Ramified primes $2, 3, 5, 17$
Class number $103680$ (GRH)
Class group $[6, 6, 12, 240]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![390625, 0, 1312500, 0, 2238125, 0, 961100, 0, 221976, 0, 24416, 0, 1581, 0, 56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 56*x^14 + 1581*x^12 + 24416*x^10 + 221976*x^8 + 961100*x^6 + 2238125*x^4 + 1312500*x^2 + 390625)
 
gp: K = bnfinit(x^16 + 56*x^14 + 1581*x^12 + 24416*x^10 + 221976*x^8 + 961100*x^6 + 2238125*x^4 + 1312500*x^2 + 390625, 1)
 

Normalized defining polynomial

\( x^{16} + 56 x^{14} + 1581 x^{12} + 24416 x^{10} + 221976 x^{8} + 961100 x^{6} + 2238125 x^{4} + 1312500 x^{2} + 390625 \)

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:  \(47991168245852798976000000000000=2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.52$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2040=2^{3}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2040}(1,·)$, $\chi_{2040}(1157,·)$, $\chi_{2040}(1223,·)$, $\chi_{2040}(1291,·)$, $\chi_{2040}(851,·)$, $\chi_{2040}(407,·)$, $\chi_{2040}(409,·)$, $\chi_{2040}(1121,·)$, $\chi_{2040}(1699,·)$, $\chi_{2040}(103,·)$, $\chi_{2040}(1259,·)$, $\chi_{2040}(1327,·)$, $\chi_{2040}(1973,·)$, $\chi_{2040}(1529,·)$, $\chi_{2040}(1597,·)$, $\chi_{2040}(373,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{25} a^{10} + \frac{6}{25} a^{8} + \frac{6}{25} a^{6} - \frac{9}{25} a^{4} + \frac{1}{25} a^{2}$, $\frac{1}{125} a^{11} + \frac{6}{125} a^{9} + \frac{31}{125} a^{7} - \frac{9}{125} a^{5} + \frac{51}{125} a^{3} + \frac{2}{5} a$, $\frac{1}{2500} a^{12} - \frac{11}{625} a^{10} + \frac{89}{625} a^{8} - \frac{296}{625} a^{6} + \frac{94}{625} a^{4} + \frac{2}{5} a^{2} + \frac{1}{4}$, $\frac{1}{12500} a^{13} - \frac{11}{3125} a^{11} + \frac{89}{3125} a^{9} - \frac{296}{3125} a^{7} + \frac{94}{3125} a^{5} - \frac{8}{25} a^{3} - \frac{3}{20} a$, $\frac{1}{457807386422562500} a^{14} - \frac{23921881233919}{457807386422562500} a^{12} - \frac{30171548522719}{6023781400296875} a^{10} + \frac{37041652009702329}{114451846605640625} a^{8} - \frac{34293066483902281}{114451846605640625} a^{6} + \frac{2238016631532}{48190251202375} a^{4} + \frac{311116568117309}{732491818276100} a^{2} - \frac{9536873685175}{29299672731044}$, $\frac{1}{2289036932112812500} a^{15} - \frac{23921881233919}{2289036932112812500} a^{13} - \frac{30171548522719}{30118907001484375} a^{11} + \frac{37041652009702329}{572259233028203125} a^{9} + \frac{80158780121738344}{572259233028203125} a^{7} + \frac{2238016631532}{240951256011875} a^{5} + \frac{1043608386393409}{3662459091380500} a^{3} - \frac{38836546416219}{146498363655220} a$

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

Class group and class number

$C_{6}\times C_{6}\times C_{12}\times C_{240}$, which has order $103680$ (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:  \( 58630.77118534252 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-170}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-255}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{6}, \sqrt{-170})\), \(\Q(\sqrt{5}, \sqrt{-34})\), \(\Q(\sqrt{30}, \sqrt{-51})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-34})\), \(\Q(\sqrt{5}, \sqrt{-51})\), \(\Q(\sqrt{30}, \sqrt{-34})\), 4.0.2312000.1, \(\Q(\zeta_{20})^+\), 4.0.5202000.2, 4.4.72000.1, 8.0.17318914560000.78, 8.0.85525504000000.22, 8.0.6927565824000000.128, 8.0.6927565824000000.82, 8.8.82944000000.2, 8.0.432972864000000.123, 8.0.27060804000000.38

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.1$x^{8} + 6 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 12$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.1$x^{8} + 6 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 12$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$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}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$