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Properties

Label 16.0.13066428858...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 13^{8}$
Root discriminant $20.88$
Ramified primes $3, 5, 13$
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, 2187, -2187, -2430, -810, -837, 594, 930, 229, -310, 66, 31, -10, 10, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 3*x^14 + 10*x^13 - 10*x^12 + 31*x^11 + 66*x^10 - 310*x^9 + 229*x^8 + 930*x^7 + 594*x^6 - 837*x^5 - 810*x^4 - 2430*x^3 - 2187*x^2 + 2187*x + 6561)
 
gp: K = bnfinit(x^16 - x^15 - 3*x^14 + 10*x^13 - 10*x^12 + 31*x^11 + 66*x^10 - 310*x^9 + 229*x^8 + 930*x^7 + 594*x^6 - 837*x^5 - 810*x^4 - 2430*x^3 - 2187*x^2 + 2187*x + 6561, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 3 x^{14} + 10 x^{13} - 10 x^{12} + 31 x^{11} + 66 x^{10} - 310 x^{9} + 229 x^{8} + 930 x^{7} + 594 x^{6} - 837 x^{5} - 810 x^{4} - 2430 x^{3} - 2187 x^{2} + 2187 x + 6561 \)

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:  \(1306642885859619140625=3^{8}\cdot 5^{12}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.88$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(195=3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{195}(64,·)$, $\chi_{195}(1,·)$, $\chi_{195}(194,·)$, $\chi_{195}(131,·)$, $\chi_{195}(77,·)$, $\chi_{195}(142,·)$, $\chi_{195}(79,·)$, $\chi_{195}(14,·)$, $\chi_{195}(92,·)$, $\chi_{195}(157,·)$, $\chi_{195}(38,·)$, $\chi_{195}(103,·)$, $\chi_{195}(116,·)$, $\chi_{195}(181,·)$, $\chi_{195}(118,·)$, $\chi_{195}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{171} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{40}{171} a^{5} + \frac{1}{3} a^{4} - \frac{4}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a - \frac{8}{19}$, $\frac{1}{513} a^{11} - \frac{1}{513} a^{10} - \frac{1}{9} a^{9} - \frac{8}{27} a^{8} + \frac{8}{27} a^{7} - \frac{131}{513} a^{6} + \frac{31}{171} a^{5} + \frac{5}{27} a^{4} - \frac{5}{27} a^{3} + \frac{4}{9} a^{2} - \frac{9}{19} a + \frac{9}{19}$, $\frac{1}{6156} a^{12} - \frac{1}{1539} a^{11} + \frac{1}{324} a^{9} + \frac{8}{81} a^{8} + \frac{238}{1539} a^{7} - \frac{1}{228} a^{6} - \frac{28}{81} a^{5} - \frac{5}{81} a^{4} + \frac{1}{4} a^{3} + \frac{74}{171} a^{2} - \frac{10}{57} a - \frac{1}{4}$, $\frac{1}{18301788} a^{13} - \frac{223}{4575447} a^{12} - \frac{1165}{1525149} a^{11} + \frac{17551}{18301788} a^{10} - \frac{29104}{240813} a^{9} + \frac{1734976}{4575447} a^{8} - \frac{2087333}{6100596} a^{7} - \frac{508438}{4575447} a^{6} - \frac{1349639}{4575447} a^{5} + \frac{124579}{321084} a^{4} + \frac{113029}{508383} a^{3} + \frac{5381}{56487} a^{2} - \frac{75347}{225948} a + \frac{2400}{18829}$, $\frac{1}{54905364} a^{14} - \frac{1}{54905364} a^{13} + \frac{959}{18301788} a^{12} + \frac{47971}{54905364} a^{11} - \frac{59491}{54905364} a^{10} + \frac{3745111}{54905364} a^{9} - \frac{1129709}{18301788} a^{8} + \frac{4566191}{54905364} a^{7} - \frac{8243519}{54905364} a^{6} + \frac{336913}{18301788} a^{5} - \frac{527239}{6100596} a^{4} - \frac{126157}{677844} a^{3} - \frac{221963}{677844} a^{2} - \frac{19689}{75316} a - \frac{15763}{75316}$, $\frac{1}{164716092} a^{15} - \frac{1}{164716092} a^{14} - \frac{1}{54905364} a^{13} - \frac{5255}{164716092} a^{12} - \frac{39619}{164716092} a^{11} + \frac{76495}{164716092} a^{10} + \frac{985333}{54905364} a^{9} - \frac{29341345}{164716092} a^{8} + \frac{3975709}{164716092} a^{7} - \frac{11706593}{54905364} a^{6} - \frac{37349}{6100596} a^{5} + \frac{1441805}{6100596} a^{4} - \frac{260639}{677844} a^{3} - \frac{212371}{677844} a^{2} + \frac{695}{75316} a - \frac{16797}{37658}$

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:  \( \frac{7}{677844} a^{15} - \frac{245}{3050298} a^{14} - \frac{7}{225948} a^{13} + \frac{35}{338922} a^{12} - \frac{35}{338922} a^{11} + \frac{217}{677844} a^{10} - \frac{11795}{3050298} a^{9} - \frac{1085}{338922} a^{8} + \frac{1603}{677844} a^{7} + \frac{1085}{112974} a^{6} + \frac{231}{37658} a^{5} - \frac{2066443}{6100596} a^{4} - \frac{315}{37658} a^{3} - \frac{945}{37658} a^{2} - \frac{1701}{75316} a + \frac{1701}{75316} \) (order $30$)
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:  \( 24866.4840857 \) (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

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-3}, \sqrt{65})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\zeta_{15})^+\), 4.4.190125.1, \(\Q(\zeta_{5})\), 4.0.21125.1, 8.0.1445900625.1, 8.8.36147515625.1, 8.0.446265625.1, \(\Q(\zeta_{15})\), 8.0.36147515625.2, 8.0.36147515625.1, 8.0.36147515625.3

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ 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
$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$5.8.6.1$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}$