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Properties

Label 20.20.1750526591...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{15}\cdot 11^{18}\cdot 661^{2}\cdot 474541^{2}$
Root discriminant $409.41$
Ramified primes $2, 5, 11, 661, 474541$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4\times C_2^4:C_5$ (as 20T75)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37203670831694647878125, 0, -3042306203259125790625, 0, 101865903320796257500, 0, -1840082858463815625, 0, 19966245005965500, 0, -136595530105375, 0, 598407347450, 0, -1660412600, 0, 2796035, 0, -2585, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2585*x^18 + 2796035*x^16 - 1660412600*x^14 + 598407347450*x^12 - 136595530105375*x^10 + 19966245005965500*x^8 - 1840082858463815625*x^6 + 101865903320796257500*x^4 - 3042306203259125790625*x^2 + 37203670831694647878125)
 
gp: K = bnfinit(x^20 - 2585*x^18 + 2796035*x^16 - 1660412600*x^14 + 598407347450*x^12 - 136595530105375*x^10 + 19966245005965500*x^8 - 1840082858463815625*x^6 + 101865903320796257500*x^4 - 3042306203259125790625*x^2 + 37203670831694647878125, 1)
 

Normalized defining polynomial

\( x^{20} - 2585 x^{18} + 2796035 x^{16} - 1660412600 x^{14} + 598407347450 x^{12} - 136595530105375 x^{10} + 19966245005965500 x^{8} - 1840082858463815625 x^{6} + 101865903320796257500 x^{4} - 3042306203259125790625 x^{2} + 37203670831694647878125 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17505265916412191723617107336923861792000000000000000=2^{20}\cdot 5^{15}\cdot 11^{18}\cdot 661^{2}\cdot 474541^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $409.41$
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:  $2, 5, 11, 661, 474541$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{275} a^{10}$, $\frac{1}{275} a^{11}$, $\frac{1}{1375} a^{12}$, $\frac{1}{1375} a^{13}$, $\frac{1}{1375} a^{14}$, $\frac{1}{1375} a^{15}$, $\frac{1}{22415881584235431875} a^{16} + \frac{349832848460177}{4483176316847086375} a^{14} - \frac{241300569862342}{896635263369417275} a^{12} + \frac{493977929129738}{896635263369417275} a^{10} - \frac{46613969441152}{3260491866797881} a^{8} - \frac{1619937321857511}{16302459333989405} a^{6} - \frac{125057366668311}{16302459333989405} a^{4} + \frac{407349581411264}{3260491866797881} a^{2} + \frac{248976802939958}{3260491866797881}$, $\frac{1}{22415881584235431875} a^{17} + \frac{349832848460177}{4483176316847086375} a^{15} - \frac{241300569862342}{896635263369417275} a^{13} + \frac{493977929129738}{896635263369417275} a^{11} - \frac{46613969441152}{3260491866797881} a^{9} - \frac{1619937321857511}{16302459333989405} a^{7} - \frac{125057366668311}{16302459333989405} a^{5} + \frac{407349581411264}{3260491866797881} a^{3} + \frac{248976802939958}{3260491866797881} a$, $\frac{1}{341156682243166661591991129821849611478999983020680320441780625} a^{18} + \frac{240803517905595691656628826460335806895682}{31014243840287878326544648165622691952636362092789120040161875} a^{16} + \frac{99213540730561293990925659135709521125904310232382101176}{563895342550688696846266330284048944593388401687074909821125} a^{14} - \frac{9664825560272948290377100263253277016616014727530753864}{563895342550688696846266330284048944593388401687074909821125} a^{12} + \frac{71396303291719390638647151940755776172276202488099103053}{112779068510137739369253266056809788918677680337414981964225} a^{10} - \frac{10593085515307389568174951342869515829799755713279760626651}{1240569753611515133061785926624907678105454483711564801606475} a^{8} + \frac{801367269786034583479642993718222138296205656440833556916}{22555813702027547873850653211361957783735536067482996392845} a^{6} + \frac{106674677270393542324616044972307909953950354509045448893}{4511162740405509574770130642272391556747107213496599278569} a^{4} - \frac{2141681869877241364818779470412482421555629834008272304713}{4511162740405509574770130642272391556747107213496599278569} a^{2} - \frac{2721376182287831693009526786491474902402449856640}{14381801623174389876532465054980835057321963978169}$, $\frac{1}{341156682243166661591991129821849611478999983020680320441780625} a^{19} + \frac{240803517905595691656628826460335806895682}{31014243840287878326544648165622691952636362092789120040161875} a^{17} + \frac{99213540730561293990925659135709521125904310232382101176}{563895342550688696846266330284048944593388401687074909821125} a^{15} - \frac{9664825560272948290377100263253277016616014727530753864}{563895342550688696846266330284048944593388401687074909821125} a^{13} + \frac{71396303291719390638647151940755776172276202488099103053}{112779068510137739369253266056809788918677680337414981964225} a^{11} - \frac{10593085515307389568174951342869515829799755713279760626651}{1240569753611515133061785926624907678105454483711564801606475} a^{9} + \frac{801367269786034583479642993718222138296205656440833556916}{22555813702027547873850653211361957783735536067482996392845} a^{7} + \frac{106674677270393542324616044972307909953950354509045448893}{4511162740405509574770130642272391556747107213496599278569} a^{5} - \frac{2141681869877241364818779470412482421555629834008272304713}{4511162740405509574770130642272391556747107213496599278569} a^{3} - \frac{2721376182287831693009526786491474902402449856640}{14381801623174389876532465054980835057321963978169} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $19$
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:  \( 22267254752000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_2^4:C_5$ (as 20T75):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_4\times C_2^4:C_5$
Character table for $C_4\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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 R $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
661Data not computed
474541Data not computed