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Properties

Label 20.0.74913657331...5137.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{9}\cdot 43^{15}$
Root discriminant $49.40$
Ramified primes $11, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![137729, 160304, 560763, 56889, 552136, 205064, 68633, 143040, 18198, -21969, 18053, -2279, -5499, 1985, 341, -591, 66, 63, -16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 16*x^18 + 63*x^17 + 66*x^16 - 591*x^15 + 341*x^14 + 1985*x^13 - 5499*x^12 - 2279*x^11 + 18053*x^10 - 21969*x^9 + 18198*x^8 + 143040*x^7 + 68633*x^6 + 205064*x^5 + 552136*x^4 + 56889*x^3 + 560763*x^2 + 160304*x + 137729)
 
gp: K = bnfinit(x^20 - 2*x^19 - 16*x^18 + 63*x^17 + 66*x^16 - 591*x^15 + 341*x^14 + 1985*x^13 - 5499*x^12 - 2279*x^11 + 18053*x^10 - 21969*x^9 + 18198*x^8 + 143040*x^7 + 68633*x^6 + 205064*x^5 + 552136*x^4 + 56889*x^3 + 560763*x^2 + 160304*x + 137729, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 16 x^{18} + 63 x^{17} + 66 x^{16} - 591 x^{15} + 341 x^{14} + 1985 x^{13} - 5499 x^{12} - 2279 x^{11} + 18053 x^{10} - 21969 x^{9} + 18198 x^{8} + 143040 x^{7} + 68633 x^{6} + 205064 x^{5} + 552136 x^{4} + 56889 x^{3} + 560763 x^{2} + 160304 x + 137729 \)

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:  \(7491365733177814925531216830515137=11^{9}\cdot 43^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.40$
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:  $11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13} a^{17} + \frac{6}{13} a^{15} + \frac{5}{13} a^{14} - \frac{4}{13} a^{12} - \frac{4}{13} a^{11} + \frac{4}{13} a^{10} + \frac{5}{13} a^{9} - \frac{3}{13} a^{8} + \frac{2}{13} a^{7} + \frac{5}{13} a^{6} + \frac{2}{13} a^{5} + \frac{1}{13} a^{4} + \frac{1}{13} a^{3} + \frac{3}{13} a^{2} - \frac{3}{13} a + \frac{4}{13}$, $\frac{1}{13} a^{18} + \frac{6}{13} a^{16} + \frac{5}{13} a^{15} - \frac{4}{13} a^{13} - \frac{4}{13} a^{12} + \frac{4}{13} a^{11} + \frac{5}{13} a^{10} - \frac{3}{13} a^{9} + \frac{2}{13} a^{8} + \frac{5}{13} a^{7} + \frac{2}{13} a^{6} + \frac{1}{13} a^{5} + \frac{1}{13} a^{4} + \frac{3}{13} a^{3} - \frac{3}{13} a^{2} + \frac{4}{13} a$, $\frac{1}{25800863139588756049811074730036079353102023438727201291} a^{19} + \frac{175138959462944874148694594497281783093200763789770467}{25800863139588756049811074730036079353102023438727201291} a^{18} - \frac{603524371600796603426178535283772516230271533442209605}{25800863139588756049811074730036079353102023438727201291} a^{17} + \frac{5733291030088879020391895466088670402137626084672484762}{25800863139588756049811074730036079353102023438727201291} a^{16} + \frac{12584063593405485435895388550342294257877249765069397664}{25800863139588756049811074730036079353102023438727201291} a^{15} - \frac{12504498792519519634331836560995904994991622915210444541}{25800863139588756049811074730036079353102023438727201291} a^{14} + \frac{1848677130580930647478370971408436087193672274768175631}{25800863139588756049811074730036079353102023438727201291} a^{13} - \frac{4214215433082435725636518161030492512445298370399153110}{25800863139588756049811074730036079353102023438727201291} a^{12} - \frac{7292461306419304872352104959336309150330162676436402108}{25800863139588756049811074730036079353102023438727201291} a^{11} - \frac{65123363673746227083277288804883227522701088802167484}{25800863139588756049811074730036079353102023438727201291} a^{10} + \frac{9669717686602109511494870248201279508199396215874756361}{25800863139588756049811074730036079353102023438727201291} a^{9} + \frac{6212783246195239004165496781981084148901683025845853710}{25800863139588756049811074730036079353102023438727201291} a^{8} - \frac{505367073090578147436198573603363156607728719749160311}{1984681779968365849985467286925852257930924879902092407} a^{7} - \frac{9791713717488684350675740554714746578702882143267597332}{25800863139588756049811074730036079353102023438727201291} a^{6} + \frac{6202849577878281906144409122715731995852899047251363758}{25800863139588756049811074730036079353102023438727201291} a^{5} - \frac{230786383114756704004728909817068521515975736753012409}{25800863139588756049811074730036079353102023438727201291} a^{4} + \frac{853957165227189028225850035905642303454247446598392842}{1984681779968365849985467286925852257930924879902092407} a^{3} + \frac{2730477682387487446545367322057508096884731628569717195}{25800863139588756049811074730036079353102023438727201291} a^{2} + \frac{6962717736108630011722670398845305170558615755263113322}{25800863139588756049811074730036079353102023438727201291} a - \frac{7110568805764463280012788564866782780925846841876258421}{25800863139588756049811074730036079353102023438727201291}$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 641704357.5703068 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-43}) \), 4.0.874577.2, 10.0.2152350613963.2

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

Sibling fields

Degree 20 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$43$43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$