↑

Properties

Label 20.4.51978079314...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{10}\cdot 5^{15}\cdot 19^{16}$
Root discriminant $61.06$
Ramified primes $3, 5, 19$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $C_2\times F_5$ (as 20T9)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-261584, 2354928, -2820992, 3529578, -2192037, -1158742, 1624938, -233197, -228717, 91658, -15441, -11134, 12491, -4929, 2026, -773, 154, -41, 9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 9*x^18 - 41*x^17 + 154*x^16 - 773*x^15 + 2026*x^14 - 4929*x^13 + 12491*x^12 - 11134*x^11 - 15441*x^10 + 91658*x^9 - 228717*x^8 - 233197*x^7 + 1624938*x^6 - 1158742*x^5 - 2192037*x^4 + 3529578*x^3 - 2820992*x^2 + 2354928*x - 261584)
 
gp: K = bnfinit(x^20 - x^19 + 9*x^18 - 41*x^17 + 154*x^16 - 773*x^15 + 2026*x^14 - 4929*x^13 + 12491*x^12 - 11134*x^11 - 15441*x^10 + 91658*x^9 - 228717*x^8 - 233197*x^7 + 1624938*x^6 - 1158742*x^5 - 2192037*x^4 + 3529578*x^3 - 2820992*x^2 + 2354928*x - 261584, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 9 x^{18} - 41 x^{17} + 154 x^{16} - 773 x^{15} + 2026 x^{14} - 4929 x^{13} + 12491 x^{12} - 11134 x^{11} - 15441 x^{10} + 91658 x^{9} - 228717 x^{8} - 233197 x^{7} + 1624938 x^{6} - 1158742 x^{5} - 2192037 x^{4} + 3529578 x^{3} - 2820992 x^{2} + 2354928 x - 261584 \)

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:  \(519780793144362253735210235595703125=3^{10}\cdot 5^{15}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.06$
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:  $3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{5} + \frac{2}{5}$, $\frac{1}{10} a^{11} - \frac{3}{10} a^{6} - \frac{3}{10} a$, $\frac{1}{50} a^{12} + \frac{1}{25} a^{10} + \frac{2}{25} a^{9} + \frac{2}{25} a^{8} - \frac{9}{50} a^{7} + \frac{7}{25} a^{6} + \frac{9}{25} a^{5} + \frac{11}{25} a^{4} + \frac{11}{25} a^{3} - \frac{11}{50} a^{2} + \frac{1}{25} a - \frac{12}{25}$, $\frac{1}{50} a^{13} + \frac{1}{25} a^{11} + \frac{2}{25} a^{10} + \frac{2}{25} a^{9} + \frac{1}{50} a^{8} - \frac{3}{25} a^{7} - \frac{6}{25} a^{6} + \frac{11}{25} a^{5} - \frac{9}{25} a^{4} - \frac{11}{50} a^{3} - \frac{9}{25} a^{2} + \frac{8}{25} a + \frac{1}{5}$, $\frac{1}{50} a^{14} - \frac{1}{50} a^{11} + \frac{3}{50} a^{9} - \frac{2}{25} a^{8} + \frac{8}{25} a^{7} + \frac{19}{50} a^{6} + \frac{3}{25} a^{5} - \frac{1}{2} a^{4} + \frac{9}{25} a^{3} + \frac{9}{25} a^{2} + \frac{1}{50} a - \frac{11}{25}$, $\frac{1}{50} a^{15} - \frac{1}{10} a^{10} - \frac{2}{5} a^{6} + \frac{23}{50} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{7}{25}$, $\frac{1}{500} a^{16} + \frac{1}{250} a^{15} - \frac{1}{125} a^{14} + \frac{1}{500} a^{13} + \frac{1}{500} a^{12} + \frac{11}{500} a^{11} - \frac{1}{125} a^{10} + \frac{13}{250} a^{9} + \frac{41}{500} a^{8} - \frac{149}{500} a^{7} - \frac{11}{500} a^{6} + \frac{53}{125} a^{5} - \frac{59}{125} a^{4} + \frac{39}{500} a^{3} + \frac{119}{500} a^{2} + \frac{7}{50} a - \frac{56}{125}$, $\frac{1}{500} a^{17} + \frac{1}{250} a^{15} - \frac{1}{500} a^{14} - \frac{1}{500} a^{13} - \frac{1}{500} a^{12} - \frac{4}{125} a^{11} - \frac{9}{125} a^{10} + \frac{19}{500} a^{9} - \frac{31}{500} a^{8} + \frac{117}{500} a^{7} + \frac{1}{125} a^{6} - \frac{7}{50} a^{5} + \frac{141}{500} a^{4} - \frac{159}{500} a^{3} + \frac{81}{250} a^{2} - \frac{97}{250} a + \frac{42}{125}$, $\frac{1}{500} a^{18} - \frac{1}{100} a^{15} - \frac{3}{500} a^{14} - \frac{3}{500} a^{13} + \frac{1}{250} a^{12} + \frac{1}{250} a^{11} - \frac{33}{500} a^{10} - \frac{33}{500} a^{9} - \frac{9}{100} a^{8} - \frac{69}{250} a^{7} - \frac{2}{125} a^{6} - \frac{183}{500} a^{5} - \frac{197}{500} a^{4} - \frac{39}{125} a^{3} + \frac{17}{125} a^{2} + \frac{27}{125} a - \frac{53}{125}$, $\frac{1}{8003689693361567058357836618656264041434159311000} a^{19} + \frac{1154126730515620732422542134618700451646201751}{1600737938672313411671567323731252808286831862200} a^{18} + \frac{6649022692022738171754360124812637439472794503}{8003689693361567058357836618656264041434159311000} a^{17} - \frac{3302387394475820957252893559464671376650984353}{8003689693361567058357836618656264041434159311000} a^{16} - \frac{22747612095387706137337744575055696565531050249}{4001844846680783529178918309328132020717079655500} a^{15} + \frac{51601781122555179439197118124980354157805375981}{8003689693361567058357836618656264041434159311000} a^{14} + \frac{7268957892632677749433809342849752394177640439}{2000922423340391764589459154664066010358539827750} a^{13} + \frac{11702033270174397332620729729506117912525508201}{8003689693361567058357836618656264041434159311000} a^{12} - \frac{149755567478424079013521456115596748696540950269}{8003689693361567058357836618656264041434159311000} a^{11} - \frac{283713001292133837939772821728362934217824375127}{4001844846680783529178918309328132020717079655500} a^{10} + \frac{440539966783154361884559603482216395853243715169}{8003689693361567058357836618656264041434159311000} a^{9} - \frac{97977553319273517349846457000879850166487723598}{1000461211670195882294729577332033005179269913875} a^{8} + \frac{628279507272642476838688835171752356260141414037}{1600737938672313411671567323731252808286831862200} a^{7} - \frac{2919018474021418777679861840902183398912554198773}{8003689693361567058357836618656264041434159311000} a^{6} - \frac{394594677854055901058185353355000732012723242609}{4001844846680783529178918309328132020717079655500} a^{5} + \frac{14730233919103608615731611007629534153529455742}{200092242334039176458945915466406601035853982775} a^{4} + \frac{3101756675674934352032328027273333792232661036029}{8003689693361567058357836618656264041434159311000} a^{3} - \frac{229027444294153624579175858404846466876914351849}{4001844846680783529178918309328132020717079655500} a^{2} + \frac{747999194826768595855700034991583913762116122999}{2000922423340391764589459154664066010358539827750} a - \frac{344078900528510136712345190875739429909133811883}{1000461211670195882294729577332033005179269913875}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( 689582360.0767176 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 5.1.16290125.1, 10.2.1326840862578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.10.8.1$x^{10} - 209 x^{5} + 11552$$5$$2$$8$$D_5$$[\ ]_{5}^{2}$
19.10.8.1$x^{10} - 209 x^{5} + 11552$$5$$2$$8$$D_5$$[\ ]_{5}^{2}$