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Properties

Label 20.6.21800592438...0000.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{40}\cdot 5^{11}\cdot 67^{8}$
Root discriminant $52.11$
Ramified primes $2, 5, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2269, -9222, 13378, -5370, 15491, -19220, -28732, 43380, -632, 3436, -5116, -5260, 7976, -2076, -300, -84, 39, 66, -22, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 22*x^18 + 66*x^17 + 39*x^16 - 84*x^15 - 300*x^14 - 2076*x^13 + 7976*x^12 - 5260*x^11 - 5116*x^10 + 3436*x^9 - 632*x^8 + 43380*x^7 - 28732*x^6 - 19220*x^5 + 15491*x^4 - 5370*x^3 + 13378*x^2 - 9222*x + 2269)
 
gp: K = bnfinit(x^20 - 2*x^19 - 22*x^18 + 66*x^17 + 39*x^16 - 84*x^15 - 300*x^14 - 2076*x^13 + 7976*x^12 - 5260*x^11 - 5116*x^10 + 3436*x^9 - 632*x^8 + 43380*x^7 - 28732*x^6 - 19220*x^5 + 15491*x^4 - 5370*x^3 + 13378*x^2 - 9222*x + 2269, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 22 x^{18} + 66 x^{17} + 39 x^{16} - 84 x^{15} - 300 x^{14} - 2076 x^{13} + 7976 x^{12} - 5260 x^{11} - 5116 x^{10} + 3436 x^{9} - 632 x^{8} + 43380 x^{7} - 28732 x^{6} - 19220 x^{5} + 15491 x^{4} - 5370 x^{3} + 13378 x^{2} - 9222 x + 2269 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-21800592438355578532659200000000000=-\,2^{40}\cdot 5^{11}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.11$
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:  $2, 5, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{10} a^{6} - \frac{1}{2} a^{5} - \frac{2}{5} a^{4} - \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{10} a^{15} + \frac{1}{10} a^{13} + \frac{1}{10} a^{10} + \frac{1}{10} a^{9} + \frac{1}{10} a^{8} - \frac{1}{2} a^{7} - \frac{2}{5} a^{6} + \frac{1}{10} a^{5} - \frac{2}{5} a^{4} - \frac{3}{10} a^{2} + \frac{1}{10} a - \frac{1}{10}$, $\frac{1}{100} a^{16} + \frac{1}{50} a^{15} + \frac{2}{25} a^{13} + \frac{9}{50} a^{12} - \frac{3}{25} a^{11} + \frac{4}{25} a^{10} + \frac{6}{25} a^{9} + \frac{1}{5} a^{8} + \frac{1}{50} a^{7} + \frac{3}{25} a^{6} + \frac{12}{25} a^{5} - \frac{7}{50} a^{4} - \frac{12}{25} a^{3} - \frac{1}{5} a^{2} + \frac{11}{25} a - \frac{1}{100}$, $\frac{1}{300} a^{17} + \frac{1}{300} a^{16} + \frac{2}{75} a^{15} + \frac{2}{75} a^{14} + \frac{1}{15} a^{13} + \frac{7}{30} a^{12} + \frac{7}{75} a^{11} - \frac{8}{75} a^{10} - \frac{11}{75} a^{9} - \frac{2}{75} a^{8} - \frac{7}{15} a^{7} + \frac{8}{25} a^{6} + \frac{4}{25} a^{5} + \frac{13}{150} a^{4} + \frac{32}{75} a^{3} - \frac{29}{75} a^{2} + \frac{1}{20} a + \frac{91}{300}$, $\frac{1}{9000} a^{18} - \frac{1}{900} a^{17} - \frac{1}{3000} a^{16} - \frac{1}{45} a^{15} + \frac{11}{4500} a^{14} - \frac{13}{75} a^{13} - \frac{431}{4500} a^{12} - \frac{43}{225} a^{11} - \frac{191}{4500} a^{10} - \frac{421}{2250} a^{9} + \frac{529}{4500} a^{8} - \frac{163}{1125} a^{7} - \frac{141}{500} a^{6} + \frac{241}{1125} a^{5} - \frac{19}{4500} a^{4} - \frac{101}{375} a^{3} + \frac{3901}{9000} a^{2} + \frac{1643}{4500} a - \frac{2471}{9000}$, $\frac{1}{404942385189531897834132343155301989000} a^{19} - \frac{414524726036967017591765346182693}{12270981369379754479822192216827333000} a^{18} - \frac{61172720949787087546761739122947153}{36812944108139263439466576650481999000} a^{17} + \frac{95390444595473892600397759107496487}{36812944108139263439466576650481999000} a^{16} + \frac{1190392415833444060504009355084402131}{202471192594765948917066171577650994500} a^{15} - \frac{8746821470207008360405591920943264909}{202471192594765948917066171577650994500} a^{14} - \frac{49207556234131300868657436394848023711}{202471192594765948917066171577650994500} a^{13} - \frac{3533939122665105916769525013992392531}{202471192594765948917066171577650994500} a^{12} - \frac{26381735304843460939156731029071452031}{202471192594765948917066171577650994500} a^{11} - \frac{18421882203218800447452579288442696603}{202471192594765948917066171577650994500} a^{10} + \frac{6117418299142613975365752032351315897}{202471192594765948917066171577650994500} a^{9} - \frac{17491578809251828091494530337636709}{523181376213865501077690365833723500} a^{8} - \frac{33025712979929445924681629599862303701}{202471192594765948917066171577650994500} a^{7} + \frac{3981080653584750080644027810637057611}{8098847703790637956682646863106039780} a^{6} + \frac{427074543075257904739954585249467683}{2699615901263545985560882287702013260} a^{5} - \frac{25222099359180097367001281140397751451}{202471192594765948917066171577650994500} a^{4} - \frac{38414261958269715596685550872305706443}{404942385189531897834132343155301989000} a^{3} + \frac{119392724957958883929111279312082452347}{404942385189531897834132343155301989000} a^{2} - \frac{33620304800505950367018033405983631899}{80988477037906379566826468631060397800} a + \frac{159490167649700073456274196551045579369}{404942385189531897834132343155301989000}$

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:  $12$
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:  \( 5573225606.27 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.5745920.1, 10.10.4126949580800000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$67$67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.6.4.1$x^{6} + 2345 x^{3} + 7756992$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
67.6.4.1$x^{6} + 2345 x^{3} + 7756992$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$