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Properties

Label 20.8.58343029654...2368.6
Degree $20$
Signature $[8, 6]$
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![-829, -8661, 25882, -20279, 17591, -13435, -10351, 9281, -11525, 9558, -4798, 4873, -1806, 1536, -757, 407, -175, -3, 3, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 3*x^18 - 3*x^17 - 175*x^16 + 407*x^15 - 757*x^14 + 1536*x^13 - 1806*x^12 + 4873*x^11 - 4798*x^10 + 9558*x^9 - 11525*x^8 + 9281*x^7 - 10351*x^6 - 13435*x^5 + 17591*x^4 - 20279*x^3 + 25882*x^2 - 8661*x - 829)
 
gp: K = bnfinit(x^20 - 4*x^19 + 3*x^18 - 3*x^17 - 175*x^16 + 407*x^15 - 757*x^14 + 1536*x^13 - 1806*x^12 + 4873*x^11 - 4798*x^10 + 9558*x^9 - 11525*x^8 + 9281*x^7 - 10351*x^6 - 13435*x^5 + 17591*x^4 - 20279*x^3 + 25882*x^2 - 8661*x - 829, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 3 x^{18} - 3 x^{17} - 175 x^{16} + 407 x^{15} - 757 x^{14} + 1536 x^{13} - 1806 x^{12} + 4873 x^{11} - 4798 x^{10} + 9558 x^{9} - 11525 x^{8} + 9281 x^{7} - 10351 x^{6} - 13435 x^{5} + 17591 x^{4} - 20279 x^{3} + 25882 x^{2} - 8661 x - 829 \)

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:  \(5834302965409512291058019417402368=2^{10}\cdot 19^{10}\cdot 43^{11}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19} a^{16} + \frac{8}{19} a^{15} + \frac{1}{19} a^{14} - \frac{4}{19} a^{13} + \frac{3}{19} a^{12} - \frac{2}{19} a^{11} - \frac{9}{19} a^{10} - \frac{9}{19} a^{9} - \frac{4}{19} a^{8} + \frac{6}{19} a^{7} - \frac{2}{19} a^{6} - \frac{5}{19} a^{5} + \frac{7}{19} a^{4} + \frac{1}{19} a^{3} - \frac{6}{19} a - \frac{8}{19}$, $\frac{1}{19} a^{17} - \frac{6}{19} a^{15} + \frac{7}{19} a^{14} - \frac{3}{19} a^{13} - \frac{7}{19} a^{12} + \frac{7}{19} a^{11} + \frac{6}{19} a^{10} - \frac{8}{19} a^{9} + \frac{7}{19} a^{7} - \frac{8}{19} a^{6} + \frac{9}{19} a^{5} + \frac{2}{19} a^{4} - \frac{8}{19} a^{3} - \frac{6}{19} a^{2} + \frac{2}{19} a + \frac{7}{19}$, $\frac{1}{95} a^{18} - \frac{2}{95} a^{17} + \frac{2}{19} a^{15} - \frac{6}{19} a^{14} + \frac{32}{95} a^{13} - \frac{18}{95} a^{12} + \frac{37}{95} a^{11} + \frac{8}{19} a^{10} + \frac{2}{95} a^{8} - \frac{43}{95} a^{7} - \frac{44}{95} a^{6} + \frac{6}{19} a^{5} + \frac{6}{19} a^{4} + \frac{7}{19} a^{3} - \frac{24}{95} a^{2} + \frac{43}{95} a + \frac{14}{95}$, $\frac{1}{554511497182116299850200561882039595492094915} a^{19} - \frac{2786165020733308882059339970696307551587734}{554511497182116299850200561882039595492094915} a^{18} + \frac{325714471309405112502326137215873494481499}{554511497182116299850200561882039595492094915} a^{17} - \frac{531964918517906471653570967751980503813500}{110902299436423259970040112376407919098418983} a^{16} - \frac{9657920841924676512911324102945590568932999}{110902299436423259970040112376407919098418983} a^{15} + \frac{189784528291974854088566486352103767205989472}{554511497182116299850200561882039595492094915} a^{14} - \frac{65903387110790303783766506513014288987161402}{554511497182116299850200561882039595492094915} a^{13} + \frac{16534122583286594431974038959299824547579303}{554511497182116299850200561882039595492094915} a^{12} + \frac{43352314669266960289825738873756929715030936}{554511497182116299850200561882039595492094915} a^{11} - \frac{39403990696595750979930189546714455946898667}{110902299436423259970040112376407919098418983} a^{10} - \frac{13034056789544573167294172694653687788367488}{554511497182116299850200561882039595492094915} a^{9} + \frac{105634114466279829232502702157671659540405898}{554511497182116299850200561882039595492094915} a^{8} + \frac{229351188588924042825327513902718106119134907}{554511497182116299850200561882039595492094915} a^{7} + \frac{73145942483094405163547075668544300156950668}{554511497182116299850200561882039595492094915} a^{6} - \frac{21094083949188027173006014116328164349078565}{110902299436423259970040112376407919098418983} a^{5} - \frac{17670670971960008499775487854175995535644}{1403826575144598227468862181979847077195177} a^{4} + \frac{21805901520141215911941214369094213228021356}{554511497182116299850200561882039595492094915} a^{3} - \frac{215829172467096313568129997830426831022120474}{554511497182116299850200561882039595492094915} a^{2} + \frac{262178788846828551484473909906960559238091618}{554511497182116299850200561882039595492094915} a + \frac{107191749898205449953635247510966190607064262}{554511497182116299850200561882039595492094915}$

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:  \( 2733375243.9 \) (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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ 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$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.8$x^{10} + x^{8} - 2 x^{6} - 2 x^{4} + x^{2} + 33$$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}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$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}$
43.4.2.1$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}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$