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Properties

Label 20.2.10242937929...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{16}\cdot 3^{18}\cdot 5^{12}\cdot 7^{17}\cdot 19^{10}\cdot 41^{5}$
Root discriminant $708.80$
Ramified primes $2, 3, 5, 7, 19, 41$
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![48728820456, 83497187880, 20689622640, -24898685040, -2480977080, -4356308556, -1202920740, 141614280, -7336320, -18873780, 8552986, -1430930, -1387860, -163440, 13290, 6495, 1365, -150, -30, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 30*x^18 - 150*x^17 + 1365*x^16 + 6495*x^15 + 13290*x^14 - 163440*x^13 - 1387860*x^12 - 1430930*x^11 + 8552986*x^10 - 18873780*x^9 - 7336320*x^8 + 141614280*x^7 - 1202920740*x^6 - 4356308556*x^5 - 2480977080*x^4 - 24898685040*x^3 + 20689622640*x^2 + 83497187880*x + 48728820456)
 
gp: K = bnfinit(x^20 - 5*x^19 - 30*x^18 - 150*x^17 + 1365*x^16 + 6495*x^15 + 13290*x^14 - 163440*x^13 - 1387860*x^12 - 1430930*x^11 + 8552986*x^10 - 18873780*x^9 - 7336320*x^8 + 141614280*x^7 - 1202920740*x^6 - 4356308556*x^5 - 2480977080*x^4 - 24898685040*x^3 + 20689622640*x^2 + 83497187880*x + 48728820456, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 30 x^{18} - 150 x^{17} + 1365 x^{16} + 6495 x^{15} + 13290 x^{14} - 163440 x^{13} - 1387860 x^{12} - 1430930 x^{11} + 8552986 x^{10} - 18873780 x^{9} - 7336320 x^{8} + 141614280 x^{7} - 1202920740 x^{6} - 4356308556 x^{5} - 2480977080 x^{4} - 24898685040 x^{3} + 20689622640 x^{2} + 83497187880 x + 48728820456 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1024293792965275270740179588611492286838179568000000000000=-\,2^{16}\cdot 3^{18}\cdot 5^{12}\cdot 7^{17}\cdot 19^{10}\cdot 41^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $708.80$
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:  $2, 3, 5, 7, 19, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{18} a^{6} + \frac{7}{18} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{36} a^{11} - \frac{1}{36} a^{10} - \frac{1}{18} a^{9} - \frac{1}{18} a^{8} + \frac{5}{36} a^{7} - \frac{5}{36} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{36} a^{12} - \frac{1}{36} a^{10} - \frac{1}{6} a^{9} - \frac{1}{36} a^{8} - \frac{1}{9} a^{7} - \frac{1}{36} a^{6} + \frac{2}{9} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a$, $\frac{1}{36} a^{13} - \frac{1}{36} a^{10} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} + \frac{1}{9} a^{7} - \frac{1}{12} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{36} a^{14} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{7}{18} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{1008} a^{15} + \frac{1}{1008} a^{14} + \frac{1}{84} a^{13} + \frac{5}{504} a^{12} + \frac{13}{1008} a^{11} - \frac{1}{48} a^{10} - \frac{29}{504} a^{9} + \frac{23}{252} a^{8} + \frac{5}{84} a^{7} - \frac{37}{252} a^{6} + \frac{17}{36} a^{5} - \frac{19}{42} a^{4} + \frac{3}{14} a^{3} - \frac{1}{7} a^{2} + \frac{5}{28} a - \frac{11}{28}$, $\frac{1}{15120} a^{16} + \frac{1}{3024} a^{15} + \frac{23}{1890} a^{14} + \frac{43}{7560} a^{13} + \frac{5}{3024} a^{12} + \frac{143}{15120} a^{11} + \frac{41}{7560} a^{10} + \frac{8}{135} a^{9} + \frac{46}{945} a^{8} - \frac{151}{1260} a^{7} + \frac{37}{1260} a^{6} - \frac{187}{630} a^{5} - \frac{1}{35} a^{4} + \frac{29}{70} a^{3} + \frac{3}{28} a^{2} - \frac{53}{140} a + \frac{8}{35}$, $\frac{1}{30240} a^{17} - \frac{1}{30240} a^{16} + \frac{1}{7560} a^{15} - \frac{41}{3780} a^{14} + \frac{229}{30240} a^{13} + \frac{173}{30240} a^{12} - \frac{103}{15120} a^{11} - \frac{323}{15120} a^{10} - \frac{7}{216} a^{9} + \frac{97}{840} a^{8} - \frac{157}{2520} a^{7} - \frac{73}{1260} a^{6} - \frac{4}{315} a^{5} + \frac{163}{420} a^{4} + \frac{341}{840} a^{3} - \frac{83}{280} a^{2} - \frac{1}{7} a - \frac{31}{140}$, $\frac{1}{30240} a^{18} - \frac{1}{30240} a^{16} - \frac{1}{2160} a^{15} + \frac{67}{6048} a^{14} - \frac{13}{2160} a^{13} - \frac{193}{30240} a^{12} + \frac{173}{15120} a^{11} - \frac{121}{7560} a^{10} - \frac{223}{7560} a^{9} + \frac{29}{1890} a^{8} - \frac{359}{2520} a^{7} - \frac{29}{180} a^{6} + \frac{104}{315} a^{5} + \frac{1}{24} a^{4} - \frac{1}{35} a^{3} + \frac{11}{40} a^{2} - \frac{1}{7} a$, $\frac{1}{94413107322607164606201815182937523597201213549934205666162224277366509313687520} a^{19} - \frac{294602226422354402706542375196523538611525078269672738634386522349444480669}{94413107322607164606201815182937523597201213549934205666162224277366509313687520} a^{18} - \frac{422210861596439588984503662599940671618324837225678107043005972447075927673}{94413107322607164606201815182937523597201213549934205666162224277366509313687520} a^{17} - \frac{2005400795008919901759000220670264865965368252646503837803478545260770760381}{94413107322607164606201815182937523597201213549934205666162224277366509313687520} a^{16} - \frac{7411927961321720837632576490769782771513389248709928364195795166706502020107}{94413107322607164606201815182937523597201213549934205666162224277366509313687520} a^{15} + \frac{815177177813341688601230492935448383802925021969919588182845540820395795672107}{94413107322607164606201815182937523597201213549934205666162224277366509313687520} a^{14} + \frac{1181612699808855143388401144874752740026596047885527760998659814388652182787669}{94413107322607164606201815182937523597201213549934205666162224277366509313687520} a^{13} - \frac{121589851087098264188890056003378020706638375633266324043045236832912916385019}{13487586760372452086600259311848217656743030507133457952308889182480929901955360} a^{12} + \frac{73890947169807957808973345804204420315266163252150314802953582412611047992177}{9441310732260716460620181518293752359720121354993420566616222427736650931368752} a^{11} + \frac{127498906085052365433475215854410998429488498007690650896850534895102568300843}{4720655366130358230310090759146876179860060677496710283308111213868325465684376} a^{10} + \frac{2550740530227479591568129230079488881748268399608760311087564623290693943064011}{23603276830651791151550453795734380899300303387483551416540556069341627328421880} a^{9} - \frac{2473799987494964690133398949688668423883917245814133231985193043967807560480309}{23603276830651791151550453795734380899300303387483551416540556069341627328421880} a^{8} + \frac{1250855559826998065039915599369404960701008860718349340129298515074016363461}{53522169684017667010318489332731022447392978202910547429797179295559245642680} a^{7} - \frac{132228249749678045941857521005433775709810555664847394205701751320978230687479}{3933879471775298525258408965955730149883383897913925236090092678223604554736980} a^{6} + \frac{228976837162134081691650433064591068226129544861664008845093321756428865199417}{874195438172288561168535325767940033307418643980872274686687261827467678830440} a^{5} - \frac{47896612775444510411893597916969666418283897670795434829970281869942000769455}{524517262903373136701121195460764019984451186388523364812012357096480607298264} a^{4} - \frac{1263063618385179090936741182171574284424514987493067263816185920608856675455897}{2622586314516865683505605977303820099922255931942616824060061785482403036491320} a^{3} - \frac{46972496268601252020412675202673137319392725909331909255874569479685359560431}{124885062596041223024076475109705719043916949140124610669526751689638239832920} a^{2} + \frac{29998390687300097278267219731112549379149264989059287150091557680129024674733}{109274429771536070146066915720992504163427330497609034335835907728433459853805} a + \frac{52521026532367993485037208323657943461897233003751873864663706506893645287}{5330459988855418055905703205902073373825723438907757772479800376996754139210}$

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:  $10$
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

\(\Q(\sqrt{57}) \), 4.2.932463.3, 5.1.388962000.3, 10.2.1123837730890952868000000.1

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 R $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
5Data not computed
$7$7.10.8.1$x^{10} - 7 x^{5} + 147$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
7.10.9.2$x^{10} + 14$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$19$19.4.2.1$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}$
19.4.2.1$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}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$