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Properties

Label 16.0.13051691536...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 13^{8}$
Root discriminant $24.11$
Ramified primes $2, 5, 13$
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![6561, 0, -5103, 0, 3240, 0, -1953, 0, 1159, 0, -217, 0, 40, 0, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^14 + 40*x^12 - 217*x^10 + 1159*x^8 - 1953*x^6 + 3240*x^4 - 5103*x^2 + 6561)
 
gp: K = bnfinit(x^16 - 7*x^14 + 40*x^12 - 217*x^10 + 1159*x^8 - 1953*x^6 + 3240*x^4 - 5103*x^2 + 6561, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{14} + 40 x^{12} - 217 x^{10} + 1159 x^{8} - 1953 x^{6} + 3240 x^{4} - 5103 x^{2} + 6561 \)

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:  \(13051691536000000000000=2^{16}\cdot 5^{12}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.11$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(260=2^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{260}(1,·)$, $\chi_{260}(131,·)$, $\chi_{260}(129,·)$, $\chi_{260}(79,·)$, $\chi_{260}(77,·)$, $\chi_{260}(207,·)$, $\chi_{260}(209,·)$, $\chi_{260}(259,·)$, $\chi_{260}(27,·)$, $\chi_{260}(157,·)$, $\chi_{260}(103,·)$, $\chi_{260}(233,·)$, $\chi_{260}(51,·)$, $\chi_{260}(53,·)$, $\chi_{260}(183,·)$, $\chi_{260}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{10431} a^{10} - \frac{1}{9} a^{8} - \frac{2}{9} a^{6} - \frac{4}{9} a^{4} + \frac{1}{9} a^{2} - \frac{217}{1159}$, $\frac{1}{31293} a^{11} - \frac{1}{27} a^{9} - \frac{2}{27} a^{7} - \frac{4}{27} a^{5} - \frac{8}{27} a^{3} + \frac{314}{1159} a$, $\frac{1}{93879} a^{12} + \frac{2}{93879} a^{10} + \frac{25}{81} a^{8} - \frac{4}{81} a^{6} - \frac{35}{81} a^{4} + \frac{314}{3477} a^{2} - \frac{177}{1159}$, $\frac{1}{281637} a^{13} + \frac{2}{281637} a^{11} + \frac{25}{243} a^{9} + \frac{77}{243} a^{7} - \frac{35}{243} a^{5} - \frac{3163}{10431} a^{3} + \frac{982}{3477} a$, $\frac{1}{844911} a^{14} + \frac{2}{844911} a^{12} - \frac{23}{844911} a^{10} + \frac{158}{729} a^{8} + \frac{127}{729} a^{6} + \frac{314}{31293} a^{4} - \frac{59}{3477} a^{2} + \frac{33}{1159}$, $\frac{1}{2534733} a^{15} + \frac{2}{2534733} a^{13} - \frac{23}{2534733} a^{11} + \frac{158}{2187} a^{9} - \frac{602}{2187} a^{7} - \frac{30979}{93879} a^{5} + \frac{3418}{10431} a^{3} - \frac{1126}{3477} a$

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:  \( \frac{679}{2534733} a^{15} - \frac{3880}{2534733} a^{13} + \frac{21049}{2534733} a^{11} - \frac{97}{2187} a^{9} + \frac{508}{2187} a^{7} - \frac{3880}{31293} a^{5} + \frac{679}{3477} a^{3} - \frac{291}{1159} a \) (order $20$)
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:  \( 51924.3982642 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{13}) \), \(\Q(i, \sqrt{65})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{-13})\), \(\Q(\sqrt{-5}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-5}, \sqrt{-13})\), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{5})\), 4.0.21125.1, 4.4.338000.1, 8.0.4569760000.1, \(\Q(\zeta_{20})\), 8.0.114244000000.2, 8.0.114244000000.1, 8.0.114244000000.3, 8.8.114244000000.1, 8.0.446265625.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.1$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}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$