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Properties

Label 20.4.19697998226...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 61^{6}\cdot 397^{6}$
Root discriminant $46.21$
Ramified primes $5, 61, 397$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T288

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-397561, 950849, -556830, 378181, 360299, -473357, 514366, -311207, 145716, -28001, -15587, 24617, -17355, 9035, -3587, 1038, -105, -74, 43, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 43*x^18 - 74*x^17 - 105*x^16 + 1038*x^15 - 3587*x^14 + 9035*x^13 - 17355*x^12 + 24617*x^11 - 15587*x^10 - 28001*x^9 + 145716*x^8 - 311207*x^7 + 514366*x^6 - 473357*x^5 + 360299*x^4 + 378181*x^3 - 556830*x^2 + 950849*x - 397561)
 
gp: K = bnfinit(x^20 - 10*x^19 + 43*x^18 - 74*x^17 - 105*x^16 + 1038*x^15 - 3587*x^14 + 9035*x^13 - 17355*x^12 + 24617*x^11 - 15587*x^10 - 28001*x^9 + 145716*x^8 - 311207*x^7 + 514366*x^6 - 473357*x^5 + 360299*x^4 + 378181*x^3 - 556830*x^2 + 950849*x - 397561, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 43 x^{18} - 74 x^{17} - 105 x^{16} + 1038 x^{15} - 3587 x^{14} + 9035 x^{13} - 17355 x^{12} + 24617 x^{11} - 15587 x^{10} - 28001 x^{9} + 145716 x^{8} - 311207 x^{7} + 514366 x^{6} - 473357 x^{5} + 360299 x^{4} + 378181 x^{3} - 556830 x^{2} + 950849 x - 397561 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1969799822613689762148183291015625=5^{10}\cdot 61^{6}\cdot 397^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.21$
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:  $5, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{2}{9} a^{8} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} - \frac{1}{9} a^{5} - \frac{1}{3} a^{4} + \frac{4}{9} a^{3} + \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{15} - \frac{2}{9} a^{11} - \frac{1}{9} a^{9} + \frac{4}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{4}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{351} a^{16} - \frac{14}{351} a^{15} + \frac{16}{351} a^{14} + \frac{47}{351} a^{13} + \frac{41}{351} a^{12} + \frac{1}{9} a^{11} + \frac{28}{351} a^{10} + \frac{34}{351} a^{9} + \frac{145}{351} a^{8} + \frac{46}{117} a^{7} - \frac{157}{351} a^{6} - \frac{58}{351} a^{5} - \frac{22}{351} a^{4} + \frac{4}{39} a^{3} + \frac{40}{117} a^{2} + \frac{109}{351} a - \frac{67}{351}$, $\frac{1}{351} a^{17} + \frac{5}{117} a^{15} - \frac{2}{351} a^{14} + \frac{4}{39} a^{13} - \frac{11}{351} a^{12} - \frac{11}{351} a^{11} + \frac{38}{117} a^{10} + \frac{4}{39} a^{9} + \frac{101}{351} a^{8} + \frac{59}{351} a^{7} + \frac{28}{117} a^{6} - \frac{2}{13} a^{5} + \frac{40}{351} a^{4} - \frac{4}{9} a^{3} - \frac{122}{351} a^{2} + \frac{172}{351} a - \frac{80}{351}$, $\frac{1}{1053} a^{18} - \frac{1}{1053} a^{16} - \frac{4}{351} a^{15} + \frac{53}{1053} a^{14} - \frac{100}{1053} a^{13} - \frac{160}{1053} a^{12} + \frac{17}{117} a^{11} + \frac{17}{1053} a^{10} + \frac{64}{1053} a^{9} + \frac{118}{1053} a^{8} - \frac{5}{39} a^{7} + \frac{313}{1053} a^{6} - \frac{241}{1053} a^{5} - \frac{38}{1053} a^{4} - \frac{425}{1053} a^{3} - \frac{461}{1053} a^{2} - \frac{49}{351} a + \frac{409}{1053}$, $\frac{1}{1259356291433877539382090692530105600584940611} a^{19} - \frac{428855339421153537580404930980208157106966}{1259356291433877539382090692530105600584940611} a^{18} + \frac{845230326575514870921217868272802358925595}{1259356291433877539382090692530105600584940611} a^{17} - \frac{1395674142166008682574830235129928980851879}{1259356291433877539382090692530105600584940611} a^{16} - \frac{42002722004595618301120923201911200881534637}{1259356291433877539382090692530105600584940611} a^{15} - \frac{10270038255138863226139015757312586607494473}{1259356291433877539382090692530105600584940611} a^{14} + \frac{94930179174471414688713384296703040673790706}{1259356291433877539382090692530105600584940611} a^{13} - \frac{89251030961457012943705941245748574963015558}{1259356291433877539382090692530105600584940611} a^{12} + \frac{114524906871955799056031732958631290069365603}{1259356291433877539382090692530105600584940611} a^{11} + \frac{4647068964720560225042342519600747088057099}{15547608536220710362741860401606241982530131} a^{10} - \frac{335627958608522785640386525029384948882885007}{1259356291433877539382090692530105600584940611} a^{9} + \frac{111732248581123879315920865893654255744817348}{1259356291433877539382090692530105600584940611} a^{8} + \frac{454153954157629501872143050287825842749474582}{1259356291433877539382090692530105600584940611} a^{7} + \frac{6505821253580800776963527870702958758483671}{15547608536220710362741860401606241982530131} a^{6} - \frac{1142493201306452934100815171285375988460581}{46642825608662131088225581204818725947590393} a^{5} + \frac{560901256343519715142110844082824964760619127}{1259356291433877539382090692530105600584940611} a^{4} + \frac{532065092629347537015330919926610652565999113}{1259356291433877539382090692530105600584940611} a^{3} - \frac{620246795144272866114337154012038530538044278}{1259356291433877539382090692530105600584940611} a^{2} - \frac{441014144803785279069453597034578655410723437}{1259356291433877539382090692530105600584940611} a - \frac{255613701294606541522523502930643193717045018}{1259356291433877539382090692530105600584940611}$

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:  $11$
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:  \( 229445959.72 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T288:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n288
Character table for t20n288 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.24217.1, 10.10.1832697153125.1, 10.2.44382426957228125.1, 10.2.14202376626313.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
397Data not computed