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Properties

Label 16.0.14096583954...4816.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 29^{8}$
Root discriminant $37.31$
Ramified primes $2, 3, 29$
Class number $54$ (GRH)
Class group $[3, 3, 6]$ (GRH)
Galois group $C_2^4$ (as 16T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5764801, 0, 0, 0, -304927, 0, 0, 0, 13728, 0, 0, 0, -127, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 127*x^12 + 13728*x^8 - 304927*x^4 + 5764801)
 
gp: K = bnfinit(x^16 - 127*x^12 + 13728*x^8 - 304927*x^4 + 5764801, 1)
 

Normalized defining polynomial

\( x^{16} - 127 x^{12} + 13728 x^{8} - 304927 x^{4} + 5764801 \)

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:  \(14096583954457373039394816=2^{32}\cdot 3^{8}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.31$
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:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(696=2^{3}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{696}(1,·)$, $\chi_{696}(581,·)$, $\chi_{696}(521,·)$, $\chi_{696}(523,·)$, $\chi_{696}(463,·)$, $\chi_{696}(407,·)$, $\chi_{696}(347,·)$, $\chi_{696}(349,·)$, $\chi_{696}(289,·)$, $\chi_{696}(233,·)$, $\chi_{696}(173,·)$, $\chi_{696}(175,·)$, $\chi_{696}(115,·)$, $\chi_{696}(695,·)$, $\chi_{696}(59,·)$, $\chi_{696}(637,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{15} a^{8} + \frac{4}{15} a^{4} + \frac{1}{15}$, $\frac{1}{105} a^{9} + \frac{34}{105} a^{5} + \frac{1}{105} a$, $\frac{1}{735} a^{10} - \frac{176}{735} a^{6} + \frac{106}{735} a^{2}$, $\frac{1}{5145} a^{11} + \frac{559}{5145} a^{7} - \frac{1364}{5145} a^{3}$, $\frac{1}{494413920} a^{12} - \frac{334}{12005} a^{8} - \frac{3201}{12005} a^{4} - \frac{2771}{41184}$, $\frac{1}{3460897440} a^{13} - \frac{334}{84035} a^{9} - \frac{3201}{84035} a^{5} - \frac{85139}{288288} a$, $\frac{1}{24226282080} a^{14} - \frac{334}{588245} a^{10} + \frac{164869}{588245} a^{6} - \frac{661715}{2018016} a^{2}$, $\frac{1}{169583974560} a^{15} - \frac{334}{4117715} a^{11} + \frac{1929604}{4117715} a^{7} - \frac{661715}{14126112} a^{3}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{6}$, which has order $54$ (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:  \( -\frac{9017}{3460897440} a^{13} + \frac{71}{252105} a^{9} - \frac{6616}{252105} a^{5} + \frac{24353}{205920} a \) (order $24$)
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:  \( 380862.253279 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4$ (as 16T3):

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 $C_2^4$
Character table for $C_2^4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-174}) \), \(\Q(\sqrt{174}) \), \(\Q(\sqrt{87}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{58}) \), \(\Q(\sqrt{-58}) \), \(\Q(\sqrt{-29}) \), \(\Q(\sqrt{29}) \), \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(i, \sqrt{174})\), \(\Q(i, \sqrt{87})\), \(\Q(i, \sqrt{58})\), \(\Q(i, \sqrt{29})\), \(\Q(\sqrt{-2}, \sqrt{87})\), \(\Q(\sqrt{-2}, \sqrt{-87})\), \(\Q(\sqrt{-2}, \sqrt{-29})\), \(\Q(\sqrt{-2}, \sqrt{29})\), \(\Q(\sqrt{2}, \sqrt{-87})\), \(\Q(\sqrt{2}, \sqrt{87})\), \(\Q(\sqrt{2}, \sqrt{29})\), \(\Q(\sqrt{2}, \sqrt{-29})\), \(\Q(\sqrt{-3}, \sqrt{58})\), \(\Q(\sqrt{-3}, \sqrt{-58})\), \(\Q(\sqrt{-3}, \sqrt{-29})\), \(\Q(\sqrt{-3}, \sqrt{29})\), \(\Q(\sqrt{3}, \sqrt{-58})\), \(\Q(\sqrt{3}, \sqrt{58})\), \(\Q(\sqrt{3}, \sqrt{29})\), \(\Q(\sqrt{3}, \sqrt{-29})\), \(\Q(\sqrt{6}, \sqrt{-29})\), \(\Q(\sqrt{6}, \sqrt{29})\), \(\Q(\sqrt{6}, \sqrt{58})\), \(\Q(\sqrt{6}, \sqrt{-58})\), \(\Q(\sqrt{-6}, \sqrt{29})\), \(\Q(\sqrt{-6}, \sqrt{-29})\), \(\Q(\sqrt{-6}, \sqrt{-58})\), \(\Q(\sqrt{-6}, \sqrt{58})\), \(\Q(\zeta_{24})\), 8.0.3754541776896.8, 8.0.46352367616.1, 8.0.3754541776896.9, 8.0.14666178816.1, 8.0.3754541776896.2, 8.0.3754541776896.3, 8.0.3754541776896.7, 8.0.234658861056.1, 8.0.3754541776896.4, 8.0.3754541776896.1, 8.0.234658861056.2, 8.0.3754541776896.6, 8.0.3754541776896.5, 8.8.3754541776896.1

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 R R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$