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Properties

Label 20.12.4650511798...5625.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 61^{8}\cdot 397^{4}$
Root discriminant $38.32$
Ramified primes $5, 61, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T794

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29, 764, -1986, -15889, 26338, -351, -60836, 103558, -82621, 18705, 30586, -35975, 17271, -2209, -2091, 1385, -375, 7, 32, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 32*x^18 + 7*x^17 - 375*x^16 + 1385*x^15 - 2091*x^14 - 2209*x^13 + 17271*x^12 - 35975*x^11 + 30586*x^10 + 18705*x^9 - 82621*x^8 + 103558*x^7 - 60836*x^6 - 351*x^5 + 26338*x^4 - 15889*x^3 - 1986*x^2 + 764*x - 29)
 
gp: K = bnfinit(x^20 - 10*x^19 + 32*x^18 + 7*x^17 - 375*x^16 + 1385*x^15 - 2091*x^14 - 2209*x^13 + 17271*x^12 - 35975*x^11 + 30586*x^10 + 18705*x^9 - 82621*x^8 + 103558*x^7 - 60836*x^6 - 351*x^5 + 26338*x^4 - 15889*x^3 - 1986*x^2 + 764*x - 29, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 32 x^{18} + 7 x^{17} - 375 x^{16} + 1385 x^{15} - 2091 x^{14} - 2209 x^{13} + 17271 x^{12} - 35975 x^{11} + 30586 x^{10} + 18705 x^{9} - 82621 x^{8} + 103558 x^{7} - 60836 x^{6} - 351 x^{5} + 26338 x^{4} - 15889 x^{3} - 1986 x^{2} + 764 x - 29 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(46505117981495597364067978515625=5^{10}\cdot 61^{8}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.32$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3635868577869047407790433420244758929335205} a^{19} - \frac{967383091416974074607348001451912319112554}{3635868577869047407790433420244758929335205} a^{18} - \frac{21940227494151068660173799191717829907887}{330533507079004309799130310931341720848655} a^{17} + \frac{115683628025841376239995417058241095309984}{727173715573809481558086684048951785867041} a^{16} + \frac{319203551704370606309707707960499139073183}{727173715573809481558086684048951785867041} a^{15} + \frac{52738551383897264596226511739243604738833}{727173715573809481558086684048951785867041} a^{14} - \frac{114443557689201599883324664848029884431246}{3635868577869047407790433420244758929335205} a^{13} - \frac{233847801732340938913847379580753761846117}{727173715573809481558086684048951785867041} a^{12} + \frac{106663667658242408192380675445203601433681}{3635868577869047407790433420244758929335205} a^{11} + \frac{886836809995004275985974852856506280700991}{3635868577869047407790433420244758929335205} a^{10} - \frac{143041555984920767207903333754704353989638}{3635868577869047407790433420244758929335205} a^{9} + \frac{898752197075464148022054989866116626508922}{3635868577869047407790433420244758929335205} a^{8} + \frac{1104039641921861316924135945013952970720326}{3635868577869047407790433420244758929335205} a^{7} + \frac{927991263675864124739266027179672700376279}{3635868577869047407790433420244758929335205} a^{6} + \frac{1492328516211993280796862467165892227187743}{3635868577869047407790433420244758929335205} a^{5} - \frac{64425111877181961973333316630892272167763}{330533507079004309799130310931341720848655} a^{4} + \frac{213518096831584238506874113965250270581817}{727173715573809481558086684048951785867041} a^{3} + \frac{1355742982913082882124873825881196493637221}{3635868577869047407790433420244758929335205} a^{2} + \frac{190682418753491886226526427193770068522242}{727173715573809481558086684048951785867041} a + \frac{1410412475427000026719744089848950685801159}{3635868577869047407790433420244758929335205}$

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:  $15$
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:  \( 567039441.142 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T794:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n794 are not computed
Character table for t20n794 is not computed

Intermediate fields

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

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{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$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$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}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.2.2$x^{4} - 61 x^{2} + 7442$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
397Data not computed