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Properties

Label 20.4.58343029654...2368.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 19^{10}\cdot 43^{11}$
Root discriminant $48.79$
Ramified primes $2, 19, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T423

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23471, 42028, -34122, -185493, 7036, 37128, -35816, 64517, -24088, 667, 9935, -7382, 2285, 220, -656, 385, -49, -47, 29, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 29*x^18 - 47*x^17 - 49*x^16 + 385*x^15 - 656*x^14 + 220*x^13 + 2285*x^12 - 7382*x^11 + 9935*x^10 + 667*x^9 - 24088*x^8 + 64517*x^7 - 35816*x^6 + 37128*x^5 + 7036*x^4 - 185493*x^3 - 34122*x^2 + 42028*x + 23471)
 
gp: K = bnfinit(x^20 - 8*x^19 + 29*x^18 - 47*x^17 - 49*x^16 + 385*x^15 - 656*x^14 + 220*x^13 + 2285*x^12 - 7382*x^11 + 9935*x^10 + 667*x^9 - 24088*x^8 + 64517*x^7 - 35816*x^6 + 37128*x^5 + 7036*x^4 - 185493*x^3 - 34122*x^2 + 42028*x + 23471, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 29 x^{18} - 47 x^{17} - 49 x^{16} + 385 x^{15} - 656 x^{14} + 220 x^{13} + 2285 x^{12} - 7382 x^{11} + 9935 x^{10} + 667 x^{9} - 24088 x^{8} + 64517 x^{7} - 35816 x^{6} + 37128 x^{5} + 7036 x^{4} - 185493 x^{3} - 34122 x^{2} + 42028 x + 23471 \)

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:  \(5834302965409512291058019417402368=2^{10}\cdot 19^{10}\cdot 43^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.79$
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:  $2, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19} a^{16} - \frac{5}{19} a^{15} - \frac{4}{19} a^{14} - \frac{9}{19} a^{12} - \frac{8}{19} a^{11} + \frac{2}{19} a^{10} + \frac{3}{19} a^{9} + \frac{9}{19} a^{8} - \frac{9}{19} a^{7} + \frac{7}{19} a^{6} - \frac{9}{19} a^{5} + \frac{5}{19} a^{4} + \frac{2}{19} a^{3} + \frac{6}{19} a^{2} - \frac{8}{19} a - \frac{2}{19}$, $\frac{1}{95} a^{17} - \frac{2}{95} a^{16} - \frac{2}{5} a^{15} - \frac{31}{95} a^{14} - \frac{28}{95} a^{13} - \frac{7}{19} a^{12} + \frac{7}{19} a^{11} + \frac{28}{95} a^{10} + \frac{18}{95} a^{9} - \frac{39}{95} a^{8} + \frac{18}{95} a^{7} + \frac{12}{95} a^{6} - \frac{22}{95} a^{5} + \frac{17}{95} a^{4} - \frac{26}{95} a^{3} + \frac{29}{95} a^{2} - \frac{7}{95} a - \frac{44}{95}$, $\frac{1}{95} a^{18} - \frac{2}{95} a^{16} - \frac{22}{95} a^{15} + \frac{7}{19} a^{14} + \frac{4}{95} a^{13} - \frac{3}{19} a^{12} - \frac{32}{95} a^{11} - \frac{36}{95} a^{10} + \frac{22}{95} a^{9} + \frac{3}{19} a^{8} - \frac{27}{95} a^{7} - \frac{3}{95} a^{6} - \frac{7}{95} a^{5} + \frac{18}{95} a^{4} - \frac{2}{5} a^{3} + \frac{6}{95} a^{2} + \frac{2}{95} a + \frac{22}{95}$, $\frac{1}{58147096463788770997749517681639563792410447420065} a^{19} - \frac{275896666560902417164802964197917134296682877926}{58147096463788770997749517681639563792410447420065} a^{18} - \frac{216368333795461806862343822118427583668003459874}{58147096463788770997749517681639563792410447420065} a^{17} + \frac{48593970137718390618134447773601585027805149814}{58147096463788770997749517681639563792410447420065} a^{16} + \frac{2329768209478698716481494354779784819165211334944}{8306728066255538713964216811662794827487206774295} a^{15} - \frac{250344008597734405614988400543001033871430758677}{8306728066255538713964216811662794827487206774295} a^{14} - \frac{12155281987783955604341813369198835077682040896128}{58147096463788770997749517681639563792410447420065} a^{13} - \frac{16981498032739063196959158784816013210828714823412}{58147096463788770997749517681639563792410447420065} a^{12} + \frac{418906804018681299387646948768749605457645243201}{58147096463788770997749517681639563792410447420065} a^{11} + \frac{1202050368960492464744222119876839435948535925502}{58147096463788770997749517681639563792410447420065} a^{10} + \frac{19350195750241604248097839039089548175388684870332}{58147096463788770997749517681639563792410447420065} a^{9} + \frac{138273343585410489429579596518079127043388084211}{1418221864970457829213402870283891799814888961465} a^{8} + \frac{8094745679538268419815273698553106749716009804143}{58147096463788770997749517681639563792410447420065} a^{7} - \frac{10753852219703805862686241768257606907691686284898}{58147096463788770997749517681639563792410447420065} a^{6} - \frac{26345271254205495220173275965835609086341152154626}{58147096463788770997749517681639563792410447420065} a^{5} - \frac{286591742932759087292542088057965520916378987565}{1661345613251107742792843362332558965497441354859} a^{4} + \frac{16379215690344890275549661277820284397641751940941}{58147096463788770997749517681639563792410447420065} a^{3} + \frac{562718684624563444600176375181765581302510169044}{8306728066255538713964216811662794827487206774295} a^{2} - \frac{14955738461675274842357233585404728447082868822201}{58147096463788770997749517681639563792410447420065} a - \frac{1649621616471500713578768941034264331066810931502}{8306728066255538713964216811662794827487206774295}$

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:  $11$
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:  \( 1117662309.57 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T423:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 160 conjugacy class representatives for t20n423 are not computed
Character table for t20n423 is not computed

Intermediate fields

\(\Q(\sqrt{817}) \), 5.5.667489.1 x5, 10.10.364007458703857.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.13$x^{10} - 15 x^{8} + 26 x^{6} - 22 x^{4} + 37 x^{2} - 59$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$