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Properties

Label 20.0.61860941990...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 11^{16}\cdot 13^{10}$
Root discriminant $54.90$
Ramified primes $5, 11, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_5\times A_4$ (as 20T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -2304, -1728, -9024, 34436, 21165, -28464, 6013, 18106, -9617, -4601, 428, 2003, 132, -353, 38, 40, -9, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 4*x^18 - 9*x^17 + 40*x^16 + 38*x^15 - 353*x^14 + 132*x^13 + 2003*x^12 + 428*x^11 - 4601*x^10 - 9617*x^9 + 18106*x^8 + 6013*x^7 - 28464*x^6 + 21165*x^5 + 34436*x^4 - 9024*x^3 - 1728*x^2 - 2304*x + 1024)
 
gp: K = bnfinit(x^20 - x^19 - 4*x^18 - 9*x^17 + 40*x^16 + 38*x^15 - 353*x^14 + 132*x^13 + 2003*x^12 + 428*x^11 - 4601*x^10 - 9617*x^9 + 18106*x^8 + 6013*x^7 - 28464*x^6 + 21165*x^5 + 34436*x^4 - 9024*x^3 - 1728*x^2 - 2304*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 4 x^{18} - 9 x^{17} + 40 x^{16} + 38 x^{15} - 353 x^{14} + 132 x^{13} + 2003 x^{12} + 428 x^{11} - 4601 x^{10} - 9617 x^{9} + 18106 x^{8} + 6013 x^{7} - 28464 x^{6} + 21165 x^{5} + 34436 x^{4} - 9024 x^{3} - 1728 x^{2} - 2304 x + 1024 \)

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:  \(61860941990830221086345856337890625=5^{10}\cdot 11^{16}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.90$
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:  $5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{17} - \frac{1}{16} a^{16} - \frac{1}{4} a^{15} + \frac{7}{16} a^{14} - \frac{1}{2} a^{13} + \frac{3}{8} a^{12} - \frac{1}{16} a^{11} + \frac{1}{4} a^{10} + \frac{3}{16} a^{9} - \frac{1}{4} a^{8} + \frac{7}{16} a^{7} - \frac{1}{16} a^{6} - \frac{3}{8} a^{5} - \frac{3}{16} a^{4} - \frac{3}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{18} - \frac{1}{64} a^{17} - \frac{1}{16} a^{16} - \frac{9}{64} a^{15} - \frac{3}{8} a^{14} - \frac{13}{32} a^{13} + \frac{31}{64} a^{12} + \frac{1}{16} a^{11} + \frac{19}{64} a^{10} - \frac{5}{16} a^{9} + \frac{7}{64} a^{8} - \frac{17}{64} a^{7} - \frac{3}{32} a^{6} - \frac{3}{64} a^{5} + \frac{1}{4} a^{4} - \frac{19}{64} a^{3} + \frac{1}{16} a^{2}$, $\frac{1}{2164325521932030267004512006381117370559877376} a^{19} - \frac{14728805522915739634164294844762128371381585}{2164325521932030267004512006381117370559877376} a^{18} + \frac{5669614440200752780239594639780291612014531}{541081380483007566751128001595279342639969344} a^{17} + \frac{162189374087527094404549250452968373460714999}{2164325521932030267004512006381117370559877376} a^{16} - \frac{37713389550401666271975331689314169386213809}{270540690241503783375564000797639671319984672} a^{15} - \frac{389049297124044970256701767721519407074503821}{1082162760966015133502256003190558685279938688} a^{14} + \frac{538239879685800014458588357121409030216784959}{2164325521932030267004512006381117370559877376} a^{13} - \frac{111062900300005938725544688183827519751421995}{541081380483007566751128001595279342639969344} a^{12} - \frac{358591043350058870707554222042026581322963245}{2164325521932030267004512006381117370559877376} a^{11} - \frac{144454346266962550218306743414843499012497649}{541081380483007566751128001595279342639969344} a^{10} - \frac{834453758646265967644089081784008428929482617}{2164325521932030267004512006381117370559877376} a^{9} - \frac{451496462019793570492799753070220760576322625}{2164325521932030267004512006381117370559877376} a^{8} - \frac{397520014138420230744993628845329607031198811}{1082162760966015133502256003190558685279938688} a^{7} + \frac{416800704894185571834555995648621678674135325}{2164325521932030267004512006381117370559877376} a^{6} + \frac{4714229228023850847317843025132899240964423}{33817586280187972921945500099704958914998084} a^{5} - \frac{464316912712700517698914493039605618493120403}{2164325521932030267004512006381117370559877376} a^{4} - \frac{25574385598644523786631696389199028330416323}{541081380483007566751128001595279342639969344} a^{3} + \frac{7880344111333073676960997160024675711740733}{33817586280187972921945500099704958914998084} a^{2} + \frac{3204346856515412407033512406687696186858595}{33817586280187972921945500099704958914998084} a - \frac{1545080046285207562823080055452733944261619}{8454396570046993230486375024926239728749521}$

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:  $9$
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:  \( 509760612.569 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times A_4$ (as 20T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 60
The 20 conjugacy class representatives for $C_5\times A_4$
Character table for $C_5\times A_4$

Intermediate fields

4.0.4225.1, \(\Q(\zeta_{11})^+\)

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

Sibling fields

Degree 30 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 $15{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ R $15{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R R $15{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ $15{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
5Data not computed
11Data not computed
13Data not computed