Properties

Label 20.0.28753215211...8032.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 193^{5}$
Root discriminant $10.54$
Ramified primes $2, 193$
Class number $1$
Class group Trivial
Galois group $D_5\wr C_2$ (as 20T48)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 6, -4, -2, 10, -10, -12, 43, -38, 21, -38, 43, -12, -10, 10, -2, -4, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 6*x^18 - 4*x^17 - 2*x^16 + 10*x^15 - 10*x^14 - 12*x^13 + 43*x^12 - 38*x^11 + 21*x^10 - 38*x^9 + 43*x^8 - 12*x^7 - 10*x^6 + 10*x^5 - 2*x^4 - 4*x^3 + 6*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 6*x^18 - 4*x^17 - 2*x^16 + 10*x^15 - 10*x^14 - 12*x^13 + 43*x^12 - 38*x^11 + 21*x^10 - 38*x^9 + 43*x^8 - 12*x^7 - 10*x^6 + 10*x^5 - 2*x^4 - 4*x^3 + 6*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 6 x^{18} - 4 x^{17} - 2 x^{16} + 10 x^{15} - 10 x^{14} - 12 x^{13} + 43 x^{12} - 38 x^{11} + 21 x^{10} - 38 x^{9} + 43 x^{8} - 12 x^{7} - 10 x^{6} + 10 x^{5} - 2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1 \)

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:  \(287532152115567788032=2^{30}\cdot 193^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.54$
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:  $2, 193$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3092351} a^{18} + \frac{814921}{3092351} a^{17} - \frac{1435426}{3092351} a^{16} - \frac{88748}{3092351} a^{15} - \frac{715639}{3092351} a^{14} + \frac{636475}{3092351} a^{13} + \frac{4222}{181903} a^{12} + \frac{1063649}{3092351} a^{11} - \frac{1172759}{3092351} a^{10} + \frac{1031245}{3092351} a^{9} - \frac{1172759}{3092351} a^{8} + \frac{1063649}{3092351} a^{7} + \frac{4222}{181903} a^{6} + \frac{636475}{3092351} a^{5} - \frac{715639}{3092351} a^{4} - \frac{88748}{3092351} a^{3} - \frac{1435426}{3092351} a^{2} + \frac{814921}{3092351} a + \frac{1}{3092351}$, $\frac{1}{3092351} a^{19} + \frac{167338}{3092351} a^{17} - \frac{371927}{3092351} a^{16} + \frac{1080432}{3092351} a^{15} + \frac{318553}{3092351} a^{14} + \frac{169178}{3092351} a^{13} - \frac{349391}{3092351} a^{12} + \frac{1090514}{3092351} a^{11} + \frac{429979}{3092351} a^{10} - \frac{886942}{3092351} a^{9} + \frac{27199}{181903} a^{8} - \frac{757304}{3092351} a^{7} - \frac{776565}{3092351} a^{6} - \frac{618235}{3092351} a^{5} - \frac{406670}{3092351} a^{4} + \frac{360645}{3092351} a^{3} + \frac{531742}{3092351} a^{2} - \frac{1489586}{3092351} a - \frac{814921}{3092351}$

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:  \( -1 \) (order $2$)
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:  \( 52.0455573525 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2$ (as 20T48):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.12352.2, 10.2.1220575232.1 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
Degree 20 siblings: data not computed
Degree 25 sibling: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$193$193.2.1.2$x^{2} + 965$$2$$1$$1$$C_2$$[\ ]_{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.1.2$x^{2} + 965$$2$$1$$1$$C_2$$[\ ]_{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.1.2$x^{2} + 965$$2$$1$$1$$C_2$$[\ ]_{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.1.2$x^{2} + 965$$2$$1$$1$$C_2$$[\ ]_{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.1.2$x^{2} + 965$$2$$1$$1$$C_2$$[\ ]_{2}$