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Properties

Label 16.0.56880090631...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 13^{12}$
Root discriminant $22.89$
Ramified primes $5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 108, 90, 21, -95, 182, 269, 262, 141, -137, 42, 13, -12, 7, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - x^14 + 7*x^13 - 12*x^12 + 13*x^11 + 42*x^10 - 137*x^9 + 141*x^8 + 262*x^7 + 269*x^6 + 182*x^5 - 95*x^4 + 21*x^3 + 90*x^2 + 108*x + 81)
 
gp: K = bnfinit(x^16 - x^15 - x^14 + 7*x^13 - 12*x^12 + 13*x^11 + 42*x^10 - 137*x^9 + 141*x^8 + 262*x^7 + 269*x^6 + 182*x^5 - 95*x^4 + 21*x^3 + 90*x^2 + 108*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - x^{14} + 7 x^{13} - 12 x^{12} + 13 x^{11} + 42 x^{10} - 137 x^{9} + 141 x^{8} + 262 x^{7} + 269 x^{6} + 182 x^{5} - 95 x^{4} + 21 x^{3} + 90 x^{2} + 108 x + 81 \)

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:  \(5688009063105712890625=5^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.89$
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:  $5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(65=5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{65}(64,·)$, $\chi_{65}(1,·)$, $\chi_{65}(8,·)$, $\chi_{65}(12,·)$, $\chi_{65}(14,·)$, $\chi_{65}(18,·)$, $\chi_{65}(21,·)$, $\chi_{65}(27,·)$, $\chi_{65}(31,·)$, $\chi_{65}(34,·)$, $\chi_{65}(38,·)$, $\chi_{65}(44,·)$, $\chi_{65}(47,·)$, $\chi_{65}(51,·)$, $\chi_{65}(53,·)$, $\chi_{65}(57,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{6} - \frac{1}{6} a$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{7} - \frac{1}{6} a^{2}$, $\frac{1}{331986} a^{13} + \frac{3178}{165993} a^{12} - \frac{3721}{331986} a^{11} - \frac{8987}{331986} a^{10} + \frac{6975}{55331} a^{9} - \frac{71747}{331986} a^{8} - \frac{7592}{55331} a^{7} + \frac{31471}{331986} a^{6} - \frac{6973}{110662} a^{5} - \frac{4495}{165993} a^{4} - \frac{101683}{331986} a^{3} - \frac{39449}{165993} a^{2} - \frac{120563}{331986} a - \frac{33801}{110662}$, $\frac{1}{995958} a^{14} - \frac{1}{995958} a^{13} - \frac{36257}{497979} a^{12} + \frac{37202}{497979} a^{11} + \frac{23539}{331986} a^{10} + \frac{140575}{995958} a^{9} - \frac{32879}{331986} a^{8} + \frac{139868}{497979} a^{7} + \frac{17582}{165993} a^{6} - \frac{485279}{995958} a^{5} + \frac{278141}{995958} a^{4} + \frac{275177}{995958} a^{3} - \frac{181408}{497979} a^{2} - \frac{13227}{55331} a + \frac{26005}{110662}$, $\frac{1}{2987874} a^{15} - \frac{1}{2987874} a^{14} - \frac{1}{2987874} a^{13} + \frac{4471}{2987874} a^{12} - \frac{3877}{995958} a^{11} - \frac{88612}{1493937} a^{10} + \frac{72791}{995958} a^{9} + \frac{574117}{2987874} a^{8} - \frac{191983}{995958} a^{7} + \frac{329545}{2987874} a^{6} - \frac{718601}{1493937} a^{5} - \frac{260161}{2987874} a^{4} - \frac{1289093}{2987874} a^{3} - \frac{250667}{995958} a^{2} - \frac{41681}{110662} a + \frac{4525}{110662}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{1763}{331986} a^{15} - \frac{1763}{331986} a^{14} + \frac{12341}{331986} a^{13} - \frac{3526}{55331} a^{12} - \frac{808}{55331} a^{11} + \frac{12341}{55331} a^{10} - \frac{241531}{331986} a^{9} + \frac{82861}{110662} a^{8} + \frac{230953}{165993} a^{7} - \frac{357312}{55331} a^{6} + \frac{160433}{165993} a^{5} - \frac{167485}{331986} a^{4} + \frac{12341}{110662} a^{3} + \frac{26445}{55331} a^{2} - \frac{574247}{165993} a + \frac{47601}{110662} \) (order $10$)
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:  \( 68071.7952641 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), 4.4.274625.1, \(\Q(\sqrt{5}, \sqrt{13})\), 4.4.274625.2, \(\Q(\zeta_{5})\), 4.0.21125.1, 4.0.54925.1, 4.0.2197.1, 8.8.75418890625.1, 8.0.446265625.1, 8.0.3016755625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
13Data not computed