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Properties

Label 20.4.31016763476...5625.2
Degree $20$
Signature $[4, 8]$
Discriminant $5^{6}\cdot 19^{8}\cdot 43^{8}$
Root discriminant $23.69$
Ramified primes $5, 19, 43$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:D_5$ (as 20T85)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, -6, 3, -24, 29, 52, 53, -50, -105, -50, 53, 52, 29, -24, 3, -6, -1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - x^18 - 6*x^17 + 3*x^16 - 24*x^15 + 29*x^14 + 52*x^13 + 53*x^12 - 50*x^11 - 105*x^10 - 50*x^9 + 53*x^8 + 52*x^7 + 29*x^6 - 24*x^5 + 3*x^4 - 6*x^3 - x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - x^18 - 6*x^17 + 3*x^16 - 24*x^15 + 29*x^14 + 52*x^13 + 53*x^12 - 50*x^11 - 105*x^10 - 50*x^9 + 53*x^8 + 52*x^7 + 29*x^6 - 24*x^5 + 3*x^4 - 6*x^3 - x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - x^{18} - 6 x^{17} + 3 x^{16} - 24 x^{15} + 29 x^{14} + 52 x^{13} + 53 x^{12} - 50 x^{11} - 105 x^{10} - 50 x^{9} + 53 x^{8} + 52 x^{7} + 29 x^{6} - 24 x^{5} + 3 x^{4} - 6 x^{3} - x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3101676347663598183510015625=5^{6}\cdot 19^{8}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.69$
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:  $5, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2}$, $\frac{1}{38} a^{12} - \frac{3}{19} a^{11} + \frac{1}{19} a^{10} - \frac{1}{19} a^{9} - \frac{2}{19} a^{8} - \frac{7}{19} a^{7} + \frac{6}{19} a^{6} - \frac{7}{19} a^{5} - \frac{2}{19} a^{4} - \frac{1}{19} a^{3} + \frac{1}{19} a^{2} + \frac{13}{38} a - \frac{9}{19}$, $\frac{1}{38} a^{13} + \frac{2}{19} a^{11} - \frac{9}{38} a^{10} + \frac{3}{38} a^{9} - \frac{1}{2} a^{8} - \frac{15}{38} a^{7} + \frac{1}{38} a^{6} + \frac{7}{38} a^{5} - \frac{7}{38} a^{4} + \frac{9}{38} a^{3} + \frac{3}{19} a^{2} + \frac{3}{38} a - \frac{13}{38}$, $\frac{1}{38} a^{14} - \frac{2}{19} a^{11} - \frac{5}{38} a^{10} - \frac{11}{38} a^{9} + \frac{1}{38} a^{8} - \frac{1}{2} a^{7} - \frac{3}{38} a^{6} + \frac{11}{38} a^{5} - \frac{13}{38} a^{4} + \frac{7}{19} a^{3} - \frac{5}{38} a^{2} + \frac{11}{38} a + \frac{15}{38}$, $\frac{1}{38} a^{15} + \frac{9}{38} a^{11} - \frac{3}{38} a^{10} - \frac{7}{38} a^{9} + \frac{3}{38} a^{8} + \frac{17}{38} a^{7} - \frac{17}{38} a^{6} + \frac{7}{38} a^{5} - \frac{1}{19} a^{4} - \frac{13}{38} a^{3} - \frac{1}{2} a^{2} - \frac{9}{38} a + \frac{2}{19}$, $\frac{1}{38} a^{16} - \frac{3}{19} a^{11} - \frac{3}{19} a^{10} + \frac{1}{19} a^{9} - \frac{2}{19} a^{8} + \frac{7}{19} a^{7} - \frac{3}{19} a^{6} - \frac{9}{38} a^{5} + \frac{2}{19} a^{4} + \frac{9}{19} a^{3} - \frac{4}{19} a^{2} - \frac{9}{19} a + \frac{5}{19}$, $\frac{1}{38} a^{17} - \frac{2}{19} a^{11} - \frac{5}{38} a^{10} + \frac{3}{38} a^{9} + \frac{9}{38} a^{8} + \frac{5}{38} a^{7} + \frac{3}{19} a^{6} + \frac{15}{38} a^{5} + \frac{13}{38} a^{4} - \frac{1}{38} a^{3} + \frac{13}{38} a^{2} - \frac{7}{38} a - \frac{13}{38}$, $\frac{1}{722} a^{18} - \frac{5}{722} a^{17} - \frac{3}{361} a^{16} - \frac{1}{361} a^{15} - \frac{2}{361} a^{14} + \frac{9}{722} a^{13} + \frac{3}{361} a^{12} + \frac{63}{722} a^{11} - \frac{9}{722} a^{10} - \frac{181}{722} a^{9} - \frac{47}{722} a^{8} + \frac{117}{361} a^{7} + \frac{60}{361} a^{6} - \frac{257}{722} a^{5} - \frac{289}{722} a^{4} - \frac{21}{722} a^{3} - \frac{60}{361} a^{2} + \frac{147}{722} a + \frac{29}{361}$, $\frac{1}{3610} a^{19} + \frac{1}{1805} a^{18} - \frac{3}{3610} a^{17} + \frac{16}{1805} a^{16} + \frac{1}{3610} a^{15} - \frac{13}{1805} a^{13} - \frac{47}{3610} a^{12} + \frac{9}{361} a^{11} - \frac{13}{361} a^{10} - \frac{101}{361} a^{9} - \frac{1}{38} a^{8} - \frac{166}{1805} a^{7} - \frac{861}{3610} a^{6} - \frac{148}{361} a^{5} - \frac{619}{3610} a^{4} + \frac{446}{1805} a^{3} + \frac{556}{1805} a^{2} - \frac{243}{3610} a - \frac{747}{1805}$

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:  $11$
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:  \( 1202623.46886 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T85):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.667489.1, 10.2.2227707825605.1, 10.6.55692695640125.1, 10.6.11138539128025.2

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 32 siblings: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$