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Properties

Label 20.8.47476467784...3125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{15}\cdot 11^{5}\cdot 9931^{5}$
Root discriminant $60.79$
Ramified primes $5, 11, 9931$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![839, 25941, 507195, 919508, -841143, -1454770, 524753, 903183, -179280, -284097, 66354, 35441, -21206, 3986, 2023, -894, -15, 18, 16, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 16*x^18 + 18*x^17 - 15*x^16 - 894*x^15 + 2023*x^14 + 3986*x^13 - 21206*x^12 + 35441*x^11 + 66354*x^10 - 284097*x^9 - 179280*x^8 + 903183*x^7 + 524753*x^6 - 1454770*x^5 - 841143*x^4 + 919508*x^3 + 507195*x^2 + 25941*x + 839)
 
gp: K = bnfinit(x^20 - 8*x^19 + 16*x^18 + 18*x^17 - 15*x^16 - 894*x^15 + 2023*x^14 + 3986*x^13 - 21206*x^12 + 35441*x^11 + 66354*x^10 - 284097*x^9 - 179280*x^8 + 903183*x^7 + 524753*x^6 - 1454770*x^5 - 841143*x^4 + 919508*x^3 + 507195*x^2 + 25941*x + 839, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 16 x^{18} + 18 x^{17} - 15 x^{16} - 894 x^{15} + 2023 x^{14} + 3986 x^{13} - 21206 x^{12} + 35441 x^{11} + 66354 x^{10} - 284097 x^{9} - 179280 x^{8} + 903183 x^{7} + 524753 x^{6} - 1454770 x^{5} - 841143 x^{4} + 919508 x^{3} + 507195 x^{2} + 25941 x + 839 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(474764677846779705681219512939453125=5^{15}\cdot 11^{5}\cdot 9931^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.79$
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:  $5, 11, 9931$
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}{431275358367514806366531326336705578858955221211335462047197} a^{19} - \frac{178384372394982725635641149157525945057733007635934920738044}{431275358367514806366531326336705578858955221211335462047197} a^{18} - \frac{131038779389931169497083480650330646843585813528226953425287}{431275358367514806366531326336705578858955221211335462047197} a^{17} + \frac{126275385231736456518416721065815886952539040959245052824924}{431275358367514806366531326336705578858955221211335462047197} a^{16} - \frac{80809273639085000786349806061232749515466424267352757116873}{431275358367514806366531326336705578858955221211335462047197} a^{15} + \frac{31508064195721068429060425671114550184585084489062642634577}{431275358367514806366531326336705578858955221211335462047197} a^{14} - \frac{131365715076502407474955057096432271914429345242855152643990}{431275358367514806366531326336705578858955221211335462047197} a^{13} + \frac{112075843863802833376535595924935141535897489926242099428185}{431275358367514806366531326336705578858955221211335462047197} a^{12} - \frac{48149442551646265950628898961347320196705069663589029906020}{431275358367514806366531326336705578858955221211335462047197} a^{11} + \frac{78112315534967038168103414147952073997467291474847023597976}{431275358367514806366531326336705578858955221211335462047197} a^{10} - \frac{66743901104556032162239520504512232432459544122814512529432}{431275358367514806366531326336705578858955221211335462047197} a^{9} + \frac{5752955289229665066339699529874646209815918007128837589297}{431275358367514806366531326336705578858955221211335462047197} a^{8} - \frac{92986104503116579571932576113941306399479858301802707007294}{431275358367514806366531326336705578858955221211335462047197} a^{7} - \frac{211746272622697327284565661913167688857589819820763873301824}{431275358367514806366531326336705578858955221211335462047197} a^{6} + \frac{1923209791313968760020745000054767356261199383863387315436}{431275358367514806366531326336705578858955221211335462047197} a^{5} + \frac{29131936922168723979372562135673559011658615858543990501847}{431275358367514806366531326336705578858955221211335462047197} a^{4} + \frac{74720094416466619960530089538657658732967644293519228129993}{431275358367514806366531326336705578858955221211335462047197} a^{3} + \frac{107120202443505312731077452454813473667193393255649932941209}{431275358367514806366531326336705578858955221211335462047197} a^{2} + \frac{105759004520816609252591978652938502581078439215880811434224}{431275358367514806366531326336705578858955221211335462047197} a - \frac{181804708157758677430223530190104604054244450323998625320766}{431275358367514806366531326336705578858955221211335462047197}$

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:  $13$
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:  \( 25521704722.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

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

Sibling fields

Degree 20 sibling: 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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
9931Data not computed