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Properties

Label 20.0.35472562029...9744.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{15}\cdot 19^{11}\cdot 43^{11}$
Root discriminant $67.22$
Ramified primes $2, 19, 43$
Class number $2268$ (GRH)
Class group $[2, 1134]$ (GRH)
Galois group 20T423

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1164241, -725088, 887494, 209487, 132457, 368252, 99143, 519267, 262887, 194067, 84784, 13337, 11112, -4747, 1885, -1180, 345, -110, 31, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 31*x^18 - 110*x^17 + 345*x^16 - 1180*x^15 + 1885*x^14 - 4747*x^13 + 11112*x^12 + 13337*x^11 + 84784*x^10 + 194067*x^9 + 262887*x^8 + 519267*x^7 + 99143*x^6 + 368252*x^5 + 132457*x^4 + 209487*x^3 + 887494*x^2 - 725088*x + 1164241)
 
gp: K = bnfinit(x^20 - 4*x^19 + 31*x^18 - 110*x^17 + 345*x^16 - 1180*x^15 + 1885*x^14 - 4747*x^13 + 11112*x^12 + 13337*x^11 + 84784*x^10 + 194067*x^9 + 262887*x^8 + 519267*x^7 + 99143*x^6 + 368252*x^5 + 132457*x^4 + 209487*x^3 + 887494*x^2 - 725088*x + 1164241, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 31 x^{18} - 110 x^{17} + 345 x^{16} - 1180 x^{15} + 1885 x^{14} - 4747 x^{13} + 11112 x^{12} + 13337 x^{11} + 84784 x^{10} + 194067 x^{9} + 262887 x^{8} + 519267 x^{7} + 99143 x^{6} + 368252 x^{5} + 132457 x^{4} + 209487 x^{3} + 887494 x^{2} - 725088 x + 1164241 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3547256202968983472963275805780639744=2^{15}\cdot 19^{11}\cdot 43^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.22$
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 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}{25473261051587584394802451661674052131955767684858079196442492495} a^{19} + \frac{6520749705148208334435755692318855161705748016606873023043403443}{25473261051587584394802451661674052131955767684858079196442492495} a^{18} - \frac{11570901400494646776392100858072553257229588472381080334248061068}{25473261051587584394802451661674052131955767684858079196442492495} a^{17} + \frac{1613987581038825912641815491993937933867597058576163631736715084}{25473261051587584394802451661674052131955767684858079196442492495} a^{16} + \frac{10504572012829535140092949878517226366137434729100535154354454023}{25473261051587584394802451661674052131955767684858079196442492495} a^{15} + \frac{1528199797323506790319270243084564806465655276357906199727960031}{25473261051587584394802451661674052131955767684858079196442492495} a^{14} - \frac{122911563006488504041958397094067469428694064052781936514812606}{1959481619352891107292496281667234779381212898835236861264807115} a^{13} - \frac{244508109865831626931150235062493458293364480048613185082726918}{25473261051587584394802451661674052131955767684858079196442492495} a^{12} + \frac{9492005119001573399746613541292877911465864653401841610101785311}{25473261051587584394802451661674052131955767684858079196442492495} a^{11} + \frac{969897054957329570344052415479470107339513127748625839057147864}{25473261051587584394802451661674052131955767684858079196442492495} a^{10} + \frac{4812771088495144263678825976738534712372354172852722670009757892}{25473261051587584394802451661674052131955767684858079196442492495} a^{9} + \frac{9154969865327323547806646931560938837231826978836924074145488581}{25473261051587584394802451661674052131955767684858079196442492495} a^{8} + \frac{10037479327702146982340185394453316310873233572855419959659021314}{25473261051587584394802451661674052131955767684858079196442492495} a^{7} - \frac{746567826565385692456733550566758053392758526003450649597650851}{5094652210317516878960490332334810426391153536971615839288498499} a^{6} - \frac{7421534019923691064139137016365347801637542408047952706803338247}{25473261051587584394802451661674052131955767684858079196442492495} a^{5} - \frac{2969927112118644578156287488884685510945012215775922250571171197}{25473261051587584394802451661674052131955767684858079196442492495} a^{4} + \frac{970091485809163726437942960116670972350795242207161114942479171}{1959481619352891107292496281667234779381212898835236861264807115} a^{3} + \frac{9962681273432982148157989445711889869864632137665747813167463963}{25473261051587584394802451661674052131955767684858079196442492495} a^{2} - \frac{1715286524405500818621119377155748565564426456361530074429350954}{5094652210317516878960490332334810426391153536971615839288498499} a - \frac{4471690754039545211158098266219905104088028248868960189582862}{23608212281360133822801160020087166016641119263075142906804905}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{1134}$, which has order $2268$ (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:  $9$
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:  \( 4747366.99665 \) (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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.15.7$x^{10} - 10 x^{8} + 56 x^{6} - 176 x^{4} + 272 x^{2} - 1184$$2$$5$$15$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 3]^{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}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$43$43.2.1.2$x^{2} + 387$$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$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$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}$