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Properties

Label 20.0.15606764239...5376.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 131^{10}$
Root discriminant $22.89$
Ramified primes $2, 131$
Class number $1$
Class group Trivial
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, -324, 0, 1769, 0, -1028, 0, 874, 0, -484, 0, 195, 0, -32, 0, 2, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 8*x^18 + 2*x^16 - 32*x^14 + 195*x^12 - 484*x^10 + 874*x^8 - 1028*x^6 + 1769*x^4 - 324*x^2 + 16)
 
gp: K = bnfinit(x^20 + 8*x^18 + 2*x^16 - 32*x^14 + 195*x^12 - 484*x^10 + 874*x^8 - 1028*x^6 + 1769*x^4 - 324*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{20} + 8 x^{18} + 2 x^{16} - 32 x^{14} + 195 x^{12} - 484 x^{10} + 874 x^{8} - 1028 x^{6} + 1769 x^{4} - 324 x^{2} + 16 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1560676423939251085562085376=2^{20}\cdot 131^{10}\)
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:  $2, 131$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{12} - \frac{1}{16} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} - \frac{15}{32} a^{4} + \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{32} a^{13} - \frac{1}{16} a^{11} - \frac{1}{32} a^{9} - \frac{1}{16} a^{7} + \frac{1}{32} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{14} + \frac{3}{32} a^{10} - \frac{1}{8} a^{8} - \frac{3}{32} a^{6} + \frac{3}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{11} - \frac{1}{8} a^{9} + \frac{1}{32} a^{7} - \frac{1}{16} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{256} a^{16} - \frac{3}{256} a^{14} + \frac{1}{128} a^{12} - \frac{19}{256} a^{10} - \frac{7}{128} a^{8} - \frac{23}{256} a^{6} + \frac{21}{256} a^{4} - \frac{13}{64} a^{2} + \frac{1}{16}$, $\frac{1}{256} a^{17} - \frac{3}{256} a^{15} + \frac{1}{128} a^{13} + \frac{13}{256} a^{11} - \frac{7}{128} a^{9} + \frac{9}{256} a^{7} + \frac{21}{256} a^{5} - \frac{5}{64} a^{3} + \frac{1}{16} a$, $\frac{1}{293625088} a^{18} - \frac{104817}{73406272} a^{16} - \frac{1806875}{293625088} a^{14} + \frac{2508995}{293625088} a^{12} - \frac{33452043}{293625088} a^{10} - \frac{11587681}{293625088} a^{8} + \frac{857607}{73406272} a^{6} + \frac{475763}{5540096} a^{4} - \frac{1526701}{3863488} a^{2} + \frac{6584311}{18351568}$, $\frac{1}{293625088} a^{19} - \frac{104817}{73406272} a^{17} - \frac{1806875}{293625088} a^{15} + \frac{2508995}{293625088} a^{13} + \frac{3251093}{293625088} a^{11} - \frac{11587681}{293625088} a^{9} - \frac{8318177}{73406272} a^{7} - \frac{909261}{5540096} a^{5} - \frac{77893}{3863488} a^{3} - \frac{2591473}{18351568} a$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{191685}{1385024} a^{19} + \frac{1552555}{1385024} a^{17} + \frac{67189}{173128} a^{15} - \frac{6082151}{1385024} a^{13} + \frac{2298797}{86564} a^{11} - \frac{89120323}{1385024} a^{9} + \frac{158456351}{1385024} a^{7} - \frac{90476387}{692512} a^{5} + \frac{263541}{1139} a^{3} - \frac{933129}{43282} a \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1283339.90144 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-131}) \), \(\Q(\sqrt{131}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{131})\), 5.1.17161.1 x5, 10.0.38579489651.1, 10.2.39505397402624.1 x5, 10.0.301567919104.3 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$131$131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$