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Properties

Label 20.20.3657076996...9477.1
Degree $20$
Signature $[20, 0]$
Discriminant $13^{11}\cdot 277^{12}$
Root discriminant $119.72$
Ramified primes $13, 277$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T138

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4818853, 18861492, 30121845, -72576179, -56577784, 103940248, 46255727, -72019254, -17656903, 26434463, 2872816, -5298560, -59616, 570887, -30938, -31258, 2871, 814, -92, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 92*x^18 + 814*x^17 + 2871*x^16 - 31258*x^15 - 30938*x^14 + 570887*x^13 - 59616*x^12 - 5298560*x^11 + 2872816*x^10 + 26434463*x^9 - 17656903*x^8 - 72019254*x^7 + 46255727*x^6 + 103940248*x^5 - 56577784*x^4 - 72576179*x^3 + 30121845*x^2 + 18861492*x - 4818853)
 
gp: K = bnfinit(x^20 - 8*x^19 - 92*x^18 + 814*x^17 + 2871*x^16 - 31258*x^15 - 30938*x^14 + 570887*x^13 - 59616*x^12 - 5298560*x^11 + 2872816*x^10 + 26434463*x^9 - 17656903*x^8 - 72019254*x^7 + 46255727*x^6 + 103940248*x^5 - 56577784*x^4 - 72576179*x^3 + 30121845*x^2 + 18861492*x - 4818853, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 92 x^{18} + 814 x^{17} + 2871 x^{16} - 31258 x^{15} - 30938 x^{14} + 570887 x^{13} - 59616 x^{12} - 5298560 x^{11} + 2872816 x^{10} + 26434463 x^{9} - 17656903 x^{8} - 72019254 x^{7} + 46255727 x^{6} + 103940248 x^{5} - 56577784 x^{4} - 72576179 x^{3} + 30121845 x^{2} + 18861492 x - 4818853 \)

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:  \(365707699675811887500665342840705936569477=13^{11}\cdot 277^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $119.72$
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:  $13, 277$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} - \frac{2}{13} a^{15} - \frac{6}{13} a^{13} - \frac{1}{13} a^{12} - \frac{4}{13} a^{11} - \frac{3}{13} a^{10} - \frac{1}{13} a^{9} + \frac{4}{13} a^{8} - \frac{3}{13} a^{7} + \frac{6}{13} a^{6} - \frac{4}{13} a^{4}$, $\frac{1}{91} a^{17} - \frac{3}{91} a^{16} + \frac{41}{91} a^{15} - \frac{32}{91} a^{14} + \frac{44}{91} a^{13} + \frac{36}{91} a^{12} + \frac{1}{91} a^{11} - \frac{37}{91} a^{10} + \frac{31}{91} a^{9} - \frac{1}{13} a^{8} + \frac{22}{91} a^{7} - \frac{32}{91} a^{6} - \frac{17}{91} a^{5} - \frac{9}{91} a^{4} + \frac{2}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{637} a^{18} + \frac{3}{637} a^{17} - \frac{5}{637} a^{16} - \frac{3}{637} a^{15} - \frac{148}{637} a^{14} - \frac{6}{49} a^{13} - \frac{4}{91} a^{12} - \frac{10}{637} a^{11} + \frac{166}{637} a^{10} + \frac{25}{637} a^{9} - \frac{41}{637} a^{8} + \frac{275}{637} a^{7} + \frac{6}{49} a^{6} - \frac{293}{637} a^{5} + \frac{149}{637} a^{4} - \frac{5}{49} a^{3} - \frac{5}{49} a^{2} + \frac{1}{49} a + \frac{2}{49}$, $\frac{1}{9648951779570849231480104884334132972714062210201521470221747487} a^{19} + \frac{3717211380337001385690394963143812816182205918928887696386338}{9648951779570849231480104884334132972714062210201521470221747487} a^{18} - \frac{26945622226548674113865249178351193807286328922765267117390203}{9648951779570849231480104884334132972714062210201521470221747487} a^{17} - \frac{16975483037293265782636706908268476548714595670827464141742998}{9648951779570849231480104884334132972714062210201521470221747487} a^{16} - \frac{1215722732853833728759272958788854719504857912452714347642957445}{9648951779570849231480104884334132972714062210201521470221747487} a^{15} - \frac{1500573305784891696444266836987071111985489298778663009549290305}{9648951779570849231480104884334132972714062210201521470221747487} a^{14} + \frac{1601226301274584956228347492026183239064556264439239492602352657}{9648951779570849231480104884334132972714062210201521470221747487} a^{13} - \frac{4335727857856730920494602679344986710065094873548083387222261622}{9648951779570849231480104884334132972714062210201521470221747487} a^{12} + \frac{1034219927030971031154191738288088522825004795364068892754870545}{9648951779570849231480104884334132972714062210201521470221747487} a^{11} - \frac{4553880999093027273490547474564182979936864322983517491450004884}{9648951779570849231480104884334132972714062210201521470221747487} a^{10} - \frac{2698406571153524281704979409797070603801694855730171788289075607}{9648951779570849231480104884334132972714062210201521470221747487} a^{9} - \frac{1534784527787694268613214908264280264706870528193520819426520785}{9648951779570849231480104884334132972714062210201521470221747487} a^{8} + \frac{3156972259841604306386794364303560590787415055451684926805381506}{9648951779570849231480104884334132972714062210201521470221747487} a^{7} + \frac{4290230653474633860548522187438212709839086125326196201598600549}{9648951779570849231480104884334132972714062210201521470221747487} a^{6} - \frac{2609698884496436502862266879185336984310229597734574589762522334}{9648951779570849231480104884334132972714062210201521470221747487} a^{5} + \frac{274346429957387886450187514537829093968742321014919424858409692}{1378421682795835604497157840619161853244866030028788781460249641} a^{4} + \frac{164902204215156413357692808337501385850045238645488822060152114}{742227059966988402421546529564164074824158631553963190017057499} a^{3} + \frac{299381603377924667734933759447363154754458028315571089523414997}{742227059966988402421546529564164074824158631553963190017057499} a^{2} + \frac{96061866586490133604139771545505801827209110817123496116536777}{742227059966988402421546529564164074824158631553963190017057499} a - \frac{92285438987373614204968302215287360042238652384991687273230248}{742227059966988402421546529564164074824158631553963190017057499}$

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:  \( 265556109352000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T138:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 28 conjugacy class representatives for t20n138
Character table for t20n138 is not computed

Intermediate fields

\(\Q(\sqrt{277}) \), 5.5.12967201.1, 10.10.46577079591509077.1

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 $20$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $20$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
277Data not computed