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Properties

Label 20.0.30606267398...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{16}\cdot 5^{15}\cdot 13^{12}$
Root discriminant $37.52$
Ramified primes $3, 5, 13$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, 583, 603, -3477, 8795, 13720, 33143, 34546, 39243, 31124, 25821, 12441, 8100, 3091, 1417, 112, 237, -44, 17, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 17*x^18 - 44*x^17 + 237*x^16 + 112*x^15 + 1417*x^14 + 3091*x^13 + 8100*x^12 + 12441*x^11 + 25821*x^10 + 31124*x^9 + 39243*x^8 + 34546*x^7 + 33143*x^6 + 13720*x^5 + 8795*x^4 - 3477*x^3 + 603*x^2 + 583*x + 121)
 
gp: K = bnfinit(x^20 - 3*x^19 + 17*x^18 - 44*x^17 + 237*x^16 + 112*x^15 + 1417*x^14 + 3091*x^13 + 8100*x^12 + 12441*x^11 + 25821*x^10 + 31124*x^9 + 39243*x^8 + 34546*x^7 + 33143*x^6 + 13720*x^5 + 8795*x^4 - 3477*x^3 + 603*x^2 + 583*x + 121, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 17 x^{18} - 44 x^{17} + 237 x^{16} + 112 x^{15} + 1417 x^{14} + 3091 x^{13} + 8100 x^{12} + 12441 x^{11} + 25821 x^{10} + 31124 x^{9} + 39243 x^{8} + 34546 x^{7} + 33143 x^{6} + 13720 x^{5} + 8795 x^{4} - 3477 x^{3} + 603 x^{2} + 583 x + 121 \)

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:  \(30606267398122877038604736328125=3^{16}\cdot 5^{15}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.52$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{10} - \frac{2}{5} a^{5} + \frac{2}{5}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{11} - \frac{2}{5} a^{6} + \frac{2}{5} a$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{23412714887599122040378276659540305253116995} a^{19} - \frac{369810402317535074186781640263360976039426}{4682542977519824408075655331908061050623399} a^{18} + \frac{2236297441053485635748274809357165701379147}{23412714887599122040378276659540305253116995} a^{17} - \frac{158127813213291347176674321620456010033637}{2128428626145374730943479696321845932101545} a^{16} + \frac{1324179131111554175001388151236753071187649}{23412714887599122040378276659540305253116995} a^{15} + \frac{1074271762500785735393373120974481413612237}{23412714887599122040378276659540305253116995} a^{14} - \frac{250990339239893669248097703125926900257312}{23412714887599122040378276659540305253116995} a^{13} + \frac{199537350960854465636541215915574086987413}{2128428626145374730943479696321845932101545} a^{12} - \frac{5425321830673682151625313115494295569782146}{23412714887599122040378276659540305253116995} a^{11} + \frac{826255753511639005887266594916597625724734}{2128428626145374730943479696321845932101545} a^{10} - \frac{8889001336664685452595164142693871853520396}{23412714887599122040378276659540305253116995} a^{9} - \frac{1560789894378331376056316809249787776257203}{4682542977519824408075655331908061050623399} a^{8} + \frac{3481818211333565094603631666436766130206476}{23412714887599122040378276659540305253116995} a^{7} + \frac{5025236447345662585219484052427598592878587}{23412714887599122040378276659540305253116995} a^{6} - \frac{751923398394875761683958269153069248639799}{2128428626145374730943479696321845932101545} a^{5} - \frac{430086325376038940054934926879883798947828}{4682542977519824408075655331908061050623399} a^{4} + \frac{1068679729415229571488476741224831729797523}{4682542977519824408075655331908061050623399} a^{3} - \frac{3549500427577271851072300147507700427438903}{23412714887599122040378276659540305253116995} a^{2} + \frac{8905916595351489853915300743966342996860589}{23412714887599122040378276659540305253116995} a + \frac{23902100778864307423400538090376778906798}{2128428626145374730943479696321845932101545}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( -\frac{44078801084538186904507227989006784617136}{23412714887599122040378276659540305253116995} a^{19} + \frac{142535153910973920262151139445549621793961}{23412714887599122040378276659540305253116995} a^{18} - \frac{776421821921753755943146707131150749719597}{23412714887599122040378276659540305253116995} a^{17} + \frac{38252624342233048856141240772402843139215}{425685725229074946188695939264369186420309} a^{16} - \frac{2167920616439283942721249410393068221943184}{4682542977519824408075655331908061050623399} a^{15} - \frac{2642864850191746886280626262909488846071887}{23412714887599122040378276659540305253116995} a^{14} - \frac{60477528088344285978051517546664518130042478}{23412714887599122040378276659540305253116995} a^{13} - \frac{10993410923285564386570016650102747400793294}{2128428626145374730943479696321845932101545} a^{12} - \frac{64055669875347322588136903543282167024368252}{4682542977519824408075655331908061050623399} a^{11} - \frac{8220407813288836632365379655869161414708982}{425685725229074946188695939264369186420309} a^{10} - \frac{979773210083082097671176901267137006217531203}{23412714887599122040378276659540305253116995} a^{9} - \frac{1057905279600043913587744354077521457886522782}{23412714887599122040378276659540305253116995} a^{8} - \frac{1316522212244622390211555735546773368995536261}{23412714887599122040378276659540305253116995} a^{7} - \frac{200115223246505614699528582679951884148775477}{4682542977519824408075655331908061050623399} a^{6} - \frac{17778886903871998520683337083718462035079985}{425685725229074946188695939264369186420309} a^{5} - \frac{147735804215545032161472547033780755665701617}{23412714887599122040378276659540305253116995} a^{4} - \frac{155307620217398872246892414690625886727047908}{23412714887599122040378276659540305253116995} a^{3} + \frac{282671426954125369903422951213593428107092511}{23412714887599122040378276659540305253116995} a^{2} - \frac{11276117674916946066536476831901564910595017}{4682542977519824408075655331908061050623399} a - \frac{383259390459898999268382436860373046748200}{425685725229074946188695939264369186420309} \) (order $10$)
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:  \( 16409998.7821 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.22244625.1, 10.10.2474116706953125.1

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 $20$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
3Data not computed
5Data not computed
13Data not computed