Properties

Label 16.0.10017750154...7776.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 13^{12}$
Root discriminant $23.72$
Ramified primes $2, 3, 13$
Class number $4$ (GRH)
Class group $[4]$ (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![81, 0, -36, 0, -146, 0, -126, 0, 327, 0, 62, 0, -9, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 3*x^14 - 9*x^12 + 62*x^10 + 327*x^8 - 126*x^6 - 146*x^4 - 36*x^2 + 81)
 
gp: K = bnfinit(x^16 + 3*x^14 - 9*x^12 + 62*x^10 + 327*x^8 - 126*x^6 - 146*x^4 - 36*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{16} + 3 x^{14} - 9 x^{12} + 62 x^{10} + 327 x^{8} - 126 x^{6} - 146 x^{4} - 36 x^{2} + 81 \)

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:  \(10017750154516748107776=2^{16}\cdot 3^{8}\cdot 13^{12}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.72$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(K));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(156=2^{2}\cdot 3\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{156}(1,·)$, $\chi_{156}(131,·)$, $\chi_{156}(5,·)$, $\chi_{156}(73,·)$, $\chi_{156}(77,·)$, $\chi_{156}(79,·)$, $\chi_{156}(83,·)$, $\chi_{156}(151,·)$, $\chi_{156}(25,·)$, $\chi_{156}(155,·)$, $\chi_{156}(31,·)$, $\chi_{156}(103,·)$, $\chi_{156}(109,·)$, $\chi_{156}(47,·)$, $\chi_{156}(53,·)$, $\chi_{156}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{2}$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{3}$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{4}$, $\frac{1}{27} a^{11} + \frac{1}{27} a^{9} + \frac{1}{9} a^{7} - \frac{2}{27} a^{5} + \frac{4}{27} a^{3}$, $\frac{1}{81} a^{12} - \frac{2}{81} a^{10} + \frac{1}{27} a^{8} - \frac{11}{81} a^{6} + \frac{10}{81} a^{4} - \frac{4}{9} a^{2}$, $\frac{1}{81} a^{13} + \frac{1}{81} a^{11} - \frac{1}{27} a^{9} - \frac{2}{81} a^{7} + \frac{4}{81} a^{5} - \frac{11}{27} a^{3}$, $\frac{1}{1042551} a^{14} - \frac{3557}{1042551} a^{12} - \frac{17879}{347517} a^{10} - \frac{6374}{1042551} a^{8} + \frac{64549}{1042551} a^{6} - \frac{9376}{38613} a^{4} - \frac{10727}{115839} a^{2} - \frac{4306}{12871}$, $\frac{1}{1042551} a^{15} - \frac{3557}{1042551} a^{13} - \frac{5008}{347517} a^{11} + \frac{32239}{1042551} a^{9} - \frac{167129}{1042551} a^{7} + \frac{5713}{347517} a^{5} - \frac{96536}{347517} a^{3} - \frac{4306}{12871} a$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( \frac{2756}{347517} a^{15} + \frac{26701}{1042551} a^{13} - \frac{64766}{1042551} a^{11} + \frac{56498}{115839} a^{9} + \frac{2775253}{1042551} a^{7} - \frac{184652}{1042551} a^{5} - \frac{9646}{115839} a^{3} - \frac{53950}{38613} a \) (order $12$)
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:  \( 46563.6853863 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{39}) \), \(\Q(\sqrt{-39}) \), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{13})\), \(\Q(i, \sqrt{39})\), \(\Q(\sqrt{3}, \sqrt{13})\), \(\Q(\sqrt{3}, \sqrt{-13})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{-13})\), 4.0.316368.2, 4.4.19773.1, 4.0.2197.1, 4.4.35152.1, 8.0.592240896.1, 8.0.100088711424.1, 8.0.1235663104.1, 8.0.100088711424.3, 8.8.100088711424.1, 8.0.100088711424.2, 8.0.390971529.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$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}$
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$