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Properties

Label 20.20.5009543363...0841.1
Degree $20$
Signature $[20, 0]$
Discriminant $29^{10}\cdot 431^{8}$
Root discriminant $60.95$
Ramified primes $29, 431$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_5$ (as 20T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7417, 126529, -729758, 1559242, -273208, -2676014, 2101035, 1396889, -1920339, -106552, 758752, -136440, -145810, 48573, 12415, -6657, -172, 396, -30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 30*x^18 + 396*x^17 - 172*x^16 - 6657*x^15 + 12415*x^14 + 48573*x^13 - 145810*x^12 - 136440*x^11 + 758752*x^10 - 106552*x^9 - 1920339*x^8 + 1396889*x^7 + 2101035*x^6 - 2676014*x^5 - 273208*x^4 + 1559242*x^3 - 729758*x^2 + 126529*x - 7417)
 
gp: K = bnfinit(x^20 - 8*x^19 - 30*x^18 + 396*x^17 - 172*x^16 - 6657*x^15 + 12415*x^14 + 48573*x^13 - 145810*x^12 - 136440*x^11 + 758752*x^10 - 106552*x^9 - 1920339*x^8 + 1396889*x^7 + 2101035*x^6 - 2676014*x^5 - 273208*x^4 + 1559242*x^3 - 729758*x^2 + 126529*x - 7417, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 30 x^{18} + 396 x^{17} - 172 x^{16} - 6657 x^{15} + 12415 x^{14} + 48573 x^{13} - 145810 x^{12} - 136440 x^{11} + 758752 x^{10} - 106552 x^{9} - 1920339 x^{8} + 1396889 x^{7} + 2101035 x^{6} - 2676014 x^{5} - 273208 x^{4} + 1559242 x^{3} - 729758 x^{2} + 126529 x - 7417 \)

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:  \(500954336335116183257628679290900841=29^{10}\cdot 431^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.95$
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:  $29, 431$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4081843430775496837189800114159480272750321} a^{19} + \frac{1839725241095329104328845352867815388486877}{4081843430775496837189800114159480272750321} a^{18} - \frac{1345646076662966091955480517929502768681905}{4081843430775496837189800114159480272750321} a^{17} + \frac{1230653879358968129068098477147903910995639}{4081843430775496837189800114159480272750321} a^{16} - \frac{1191838147804876218925013250051940204034328}{4081843430775496837189800114159480272750321} a^{15} - \frac{447649902761215674031530592434650731322759}{4081843430775496837189800114159480272750321} a^{14} + \frac{572902354937004009354274344677956059250759}{4081843430775496837189800114159480272750321} a^{13} + \frac{745840807705131826482047635638669913865726}{4081843430775496837189800114159480272750321} a^{12} - \frac{1671802336436061784010959232146621032304045}{4081843430775496837189800114159480272750321} a^{11} - \frac{868116575776481133636438894530368682820774}{4081843430775496837189800114159480272750321} a^{10} - \frac{184697559001811369069420098736835522255564}{4081843430775496837189800114159480272750321} a^{9} - \frac{1307943643302542595971484751900266363396087}{4081843430775496837189800114159480272750321} a^{8} - \frac{1311937907191258404103439988172643277655248}{4081843430775496837189800114159480272750321} a^{7} + \frac{957167930010769066293263549459093853885457}{4081843430775496837189800114159480272750321} a^{6} - \frac{931767364292498892137535568130957303504145}{4081843430775496837189800114159480272750321} a^{5} + \frac{1925714313818288194359042167816552401122135}{4081843430775496837189800114159480272750321} a^{4} + \frac{1200882457855225337941228024586409947701566}{4081843430775496837189800114159480272750321} a^{3} - \frac{1784133298642273195017395888525794197860439}{4081843430775496837189800114159480272750321} a^{2} - \frac{680584212244507811871734500521860809747590}{4081843430775496837189800114159480272750321} a + \frac{175257648452335961333965084345728962560856}{4081843430775496837189800114159480272750321}$

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:  $19$
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:  \( 479833200273 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 20T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

\(\Q(\sqrt{29}) \), 10.10.841594859912069.1

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

Sibling fields

Degree 5 sibling: data not computed
Degree 6 sibling: data not computed
Degree 10 siblings: data not computed
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
431Data not computed