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Properties

Label 20.4.31777145476...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{15}\cdot 19^{18}$
Root discriminant $47.33$
Ramified primes $5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-119475, -35775, 21600, 320775, -249815, 184205, -200155, 151255, -54804, 50437, -29195, 7230, -5209, 2555, 116, 380, -14, -20, -10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 10*x^18 - 20*x^17 - 14*x^16 + 380*x^15 + 116*x^14 + 2555*x^13 - 5209*x^12 + 7230*x^11 - 29195*x^10 + 50437*x^9 - 54804*x^8 + 151255*x^7 - 200155*x^6 + 184205*x^5 - 249815*x^4 + 320775*x^3 + 21600*x^2 - 35775*x - 119475)
 
gp: K = bnfinit(x^20 - 2*x^19 - 10*x^18 - 20*x^17 - 14*x^16 + 380*x^15 + 116*x^14 + 2555*x^13 - 5209*x^12 + 7230*x^11 - 29195*x^10 + 50437*x^9 - 54804*x^8 + 151255*x^7 - 200155*x^6 + 184205*x^5 - 249815*x^4 + 320775*x^3 + 21600*x^2 - 35775*x - 119475, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 10 x^{18} - 20 x^{17} - 14 x^{16} + 380 x^{15} + 116 x^{14} + 2555 x^{13} - 5209 x^{12} + 7230 x^{11} - 29195 x^{10} + 50437 x^{9} - 54804 x^{8} + 151255 x^{7} - 200155 x^{6} + 184205 x^{5} - 249815 x^{4} + 320775 x^{3} + 21600 x^{2} - 35775 x - 119475 \)

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:  \(3177714547665748337794219970703125=5^{15}\cdot 19^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.33$
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:  $5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{14} + \frac{1}{10} a^{13} + \frac{1}{5} a^{12} - \frac{1}{2} a^{11} - \frac{1}{10} a^{10} + \frac{1}{5} a^{9} + \frac{3}{10} a^{8} - \frac{3}{10} a^{7} + \frac{2}{5} a^{6} - \frac{1}{2} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{15} + \frac{1}{10} a^{13} + \frac{3}{10} a^{12} + \frac{2}{5} a^{11} + \frac{3}{10} a^{10} + \frac{1}{10} a^{9} + \frac{2}{5} a^{8} - \frac{3}{10} a^{7} + \frac{1}{10} a^{6} - \frac{2}{5} a^{5} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{16} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a$, $\frac{1}{30} a^{17} + \frac{1}{30} a^{16} - \frac{1}{30} a^{15} + \frac{1}{30} a^{14} + \frac{7}{30} a^{13} - \frac{1}{3} a^{12} + \frac{11}{30} a^{11} - \frac{13}{30} a^{10} + \frac{4}{15} a^{9} - \frac{3}{10} a^{8} + \frac{7}{30} a^{7} - \frac{7}{15} a^{6} - \frac{1}{10} a^{5} - \frac{11}{30} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{900} a^{18} - \frac{2}{225} a^{17} - \frac{7}{900} a^{16} + \frac{11}{450} a^{15} - \frac{11}{900} a^{14} + \frac{44}{225} a^{13} + \frac{19}{180} a^{12} + \frac{73}{180} a^{11} - \frac{167}{450} a^{10} - \frac{19}{100} a^{9} + \frac{181}{900} a^{8} - \frac{7}{450} a^{7} + \frac{3}{20} a^{6} - \frac{61}{180} a^{5} + \frac{13}{45} a^{4} - \frac{37}{90} a^{3} - \frac{11}{36} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{1320000968592169160178910751650531837317331629480600} a^{19} - \frac{625753038158522881966294643454674414175565814111}{1320000968592169160178910751650531837317331629480600} a^{18} + \frac{11982987714172509347345420553787447185860553623897}{1320000968592169160178910751650531837317331629480600} a^{17} - \frac{41831399223069709364395038518249238175020051130517}{1320000968592169160178910751650531837317331629480600} a^{16} + \frac{3182548634111205048758050411171545287947835010763}{1320000968592169160178910751650531837317331629480600} a^{15} + \frac{385366876068088697137434777091061777477728687189}{32195145575418760004363676869525166763837356816600} a^{14} - \frac{66115933912806239421520906575297832436584186565173}{1320000968592169160178910751650531837317331629480600} a^{13} + \frac{9532665666222349974737491164286648089272963113367}{33000024214804229004472768791263295932933290737015} a^{12} + \frac{9450002024112078489878315936382428714748333880171}{32195145575418760004363676869525166763837356816600} a^{11} + \frac{195113206982696214070142758968100595258807938644437}{440000322864056386726303583883510612439110543160200} a^{10} + \frac{292448816655807289659464570441944081802324336930147}{660000484296084580089455375825265918658665814740300} a^{9} + \frac{113333321175731430194754593931087077204792598601723}{1320000968592169160178910751650531837317331629480600} a^{8} - \frac{100126932882031359704000721018308310576965093796521}{440000322864056386726303583883510612439110543160200} a^{7} + \frac{7172315035268278240114532284623220700930401986917}{26400019371843383203578215033010636746346632589612} a^{6} + \frac{96853370135043156051546828994313176102468961277259}{264000193718433832035782150330106367463466325896120} a^{5} - \frac{3621179869144064881237166293361137232729242297475}{26400019371843383203578215033010636746346632589612} a^{4} + \frac{70955481130740205412004077683929952140676735121927}{264000193718433832035782150330106367463466325896120} a^{3} + \frac{508456603461207877996877334467899259297570445259}{4400003228640563867263035838835106124391105431602} a^{2} - \frac{346739820764071175593097034995133073682582256860}{733333871440093977877172639805851020731850905267} a - \frac{716301947487317211738276736258783484695605238545}{1955556990506917274339127039482269388618269080712}$

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:  \( 2565467606.734227 \) (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

\(\Q(\sqrt{5}) \), 4.4.45125.1, 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$19$19.10.9.1$x^{10} - 19$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$
19.10.9.1$x^{10} - 19$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$