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Properties

Label 20.0.60885938361...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 3^{16}\cdot 5^{38}\cdot 11^{10}\cdot 89^{5}$
Root discriminant $1377.93$
Ramified primes $2, 3, 5, 11, 89$
Class number Not computed
Class group Not computed
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9480619878400, -3924318464000, 410982528000, -477334528000, 636643200000, -125420828160, -50024230400, 15243180800, 266531200, -106188800, -16160544, -19560160, 5426320, 213280, -117440, -18672, 3960, 280, -30, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 - 30*x^18 + 280*x^17 + 3960*x^16 - 18672*x^15 - 117440*x^14 + 213280*x^13 + 5426320*x^12 - 19560160*x^11 - 16160544*x^10 - 106188800*x^9 + 266531200*x^8 + 15243180800*x^7 - 50024230400*x^6 - 125420828160*x^5 + 636643200000*x^4 - 477334528000*x^3 + 410982528000*x^2 - 3924318464000*x + 9480619878400)
 
gp: K = bnfinit(x^20 - 10*x^19 - 30*x^18 + 280*x^17 + 3960*x^16 - 18672*x^15 - 117440*x^14 + 213280*x^13 + 5426320*x^12 - 19560160*x^11 - 16160544*x^10 - 106188800*x^9 + 266531200*x^8 + 15243180800*x^7 - 50024230400*x^6 - 125420828160*x^5 + 636643200000*x^4 - 477334528000*x^3 + 410982528000*x^2 - 3924318464000*x + 9480619878400, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} - 30 x^{18} + 280 x^{17} + 3960 x^{16} - 18672 x^{15} - 117440 x^{14} + 213280 x^{13} + 5426320 x^{12} - 19560160 x^{11} - 16160544 x^{10} - 106188800 x^{9} + 266531200 x^{8} + 15243180800 x^{7} - 50024230400 x^{6} - 125420828160 x^{5} + 636643200000 x^{4} - 477334528000 x^{3} + 410982528000 x^{2} - 3924318464000 x + 9480619878400 \)

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:  \(608859383615347881779750989160156250000000000000000000000000000=2^{28}\cdot 3^{16}\cdot 5^{38}\cdot 11^{10}\cdot 89^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1377.93$
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:  $2, 3, 5, 11, 89$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{800} a^{10} + \frac{1}{40} a^{9} + \frac{1}{80} a^{8} + \frac{1}{20} a^{7} - \frac{1}{40} a^{6} + \frac{2}{25} a^{5} - \frac{2}{5}$, $\frac{1}{1600} a^{11} + \frac{1}{160} a^{9} + \frac{1}{40} a^{8} - \frac{1}{80} a^{7} + \frac{1}{25} a^{6} + \frac{3}{40} a^{5} - \frac{1}{5} a$, $\frac{1}{1600} a^{12} + \frac{1}{40} a^{9} - \frac{1}{80} a^{8} + \frac{1}{25} a^{7} - \frac{1}{20} a^{6} + \frac{1}{10} a^{5} - \frac{1}{5} a^{2}$, $\frac{1}{3200} a^{13} + \frac{1}{40} a^{9} + \frac{1}{50} a^{8} - \frac{1}{40} a^{7} + \frac{1}{20} a^{6} + \frac{3}{40} a^{5} - \frac{1}{10} a^{3}$, $\frac{1}{16000} a^{14} + \frac{1}{16000} a^{13} - \frac{1}{4000} a^{12} - \frac{1}{4000} a^{11} - \frac{1}{4000} a^{10} - \frac{21}{1000} a^{9} - \frac{17}{2000} a^{8} - \frac{41}{1000} a^{7} - \frac{41}{1000} a^{6} + \frac{9}{1000} a^{5} - \frac{7}{100} a^{4} + \frac{2}{25} a^{3} + \frac{9}{50} a^{2} - \frac{8}{25} a - \frac{8}{25}$, $\frac{1}{32000} a^{15} + \frac{1}{4000} a^{10} + \frac{1}{80} a^{9} - \frac{1}{40} a^{8} + \frac{3}{80} a^{7} + \frac{1}{40} a^{6} - \frac{9}{125} a^{5} - \frac{1}{20} a^{4} - \frac{6}{25}$, $\frac{1}{704000} a^{16} - \frac{1}{176000} a^{15} + \frac{7}{352000} a^{14} + \frac{1}{176000} a^{13} + \frac{23}{88000} a^{12} - \frac{3}{44000} a^{11} + \frac{1}{22000} a^{10} - \frac{93}{5500} a^{9} - \frac{859}{44000} a^{8} - \frac{1007}{22000} a^{7} - \frac{119}{2000} a^{6} - \frac{173}{2750} a^{5} - \frac{7}{100} a^{4} - \frac{117}{1100} a^{3} + \frac{9}{550} a^{2} - \frac{17}{50} a + \frac{4}{25}$, $\frac{1}{1408000} a^{17} - \frac{1}{704000} a^{15} - \frac{7}{352000} a^{14} + \frac{7}{88000} a^{13} - \frac{7}{35200} a^{12} + \frac{3}{22000} a^{11} + \frac{1}{88000} a^{10} + \frac{763}{88000} a^{9} + \frac{337}{22000} a^{8} + \frac{2627}{44000} a^{7} + \frac{1167}{22000} a^{6} + \frac{827}{11000} a^{5} - \frac{271}{2200} a^{4} - \frac{19}{550} a^{3} - \frac{19}{1100} a^{2} + \frac{11}{50} a + \frac{1}{25}$, $\frac{1}{1408000} a^{18} + \frac{1}{176000} a^{15} - \frac{9}{352000} a^{14} - \frac{1}{176000} a^{13} + \frac{3}{11000} a^{12} - \frac{1}{5500} a^{11} - \frac{47}{88000} a^{10} - \frac{1247}{44000} a^{9} + \frac{3}{1375} a^{8} + \frac{271}{5500} a^{7} + \frac{169}{22000} a^{6} - \frac{177}{11000} a^{5} - \frac{4}{275} a^{4} + \frac{32}{275} a^{3} + \frac{21}{275} a^{2} - \frac{23}{50} a - \frac{6}{25}$, $\frac{1}{48651493943851811808960744823566226955562884036241367455507063566080000} a^{19} + \frac{2243915453301293839572292866905089034435281789177792338003251}{6910723571569859631954651253347475419824273300602466968111798802000} a^{18} + \frac{824038868582787867467277859394074322063352516558753563917145257}{2432574697192590590448037241178311347778144201812068372775353178304000} a^{17} + \frac{987989323624826561497351003253494895483040129100394026303260321}{2432574697192590590448037241178311347778144201812068372775353178304000} a^{16} + \frac{2786251183383139393974785357611006856675523225058938760148433853}{221143154290235508222548840107119213434376745619278942979577561664000} a^{15} - \frac{30024981599596700774197215771831382607467173528515958632744493251}{1520359185745369119030023275736444592361340126132542732984595736440000} a^{14} - \frac{21122813678522904129489588004541603361446635702044053439785206761}{608143674298147647612009310294577836944536050453017093193838294576000} a^{13} - \frac{1347269962380041007577440344376298248653631683411990261851556543}{7601795928726845595150116378682222961806700630662713664922978682200} a^{12} - \frac{21241853017260085000013835106998388175894322950268711438514339133}{121628734859629529522401862058915567388907210090603418638767658915200} a^{11} + \frac{7126362872201778103839442923799915664166968688722575331509196807}{12162873485962952952240186205891556738890721009060341863876765891520} a^{10} + \frac{6312497786401056781392448052985060482376365681044210608401771534087}{380089796436342279757505818934111148090335031533135683246148934110000} a^{9} - \frac{1001454155364926411709351322639242369997189410652980092251537090343}{152035918574536911903002327573644459236134012613254273298459573644000} a^{8} - \frac{5767167628527524454722228389888815317414551411522146917698885764507}{152035918574536911903002327573644459236134012613254273298459573644000} a^{7} - \frac{73476114019800146591150137911646535627028470451673723132909664617}{1727680892892464907988662813336868854956068325150616742027949700500} a^{6} + \frac{955339124635435899391174214061366026574514779737624890959051096419}{19004489821817113987875290946705557404516751576656784162307446705500} a^{5} + \frac{72503472383862729893293050749983182549000168923026515023202953141}{863840446446232453994331406668434427478034162575308371013974850250} a^{4} + \frac{584331950868031332760342146346679145634029661830945295696288367151}{3800897964363422797575058189341111480903350315331356832461489341100} a^{3} + \frac{422448802937253736330681924265264072967185573377111436161504358729}{3800897964363422797575058189341111480903350315331356832461489341100} a^{2} - \frac{51378453377448961030260103221345998042252037738963091290171628979}{172768089289246490798866281333686885495606832515061674202794970050} a - \frac{7028432996385567066378289363060659026294565526859127816658430176}{86384044644623245399433140666843442747803416257530837101397485025}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{55}) \), 4.0.4307600.1, 5.1.2531250000.22, 10.2.330204878437500000000000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.10.19.1$x^{10} + 5$$10$$1$$19$$F_5$$[9/4]_{4}$
5.10.19.1$x^{10} + 5$$10$$1$$19$$F_5$$[9/4]_{4}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$