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Properties

Label 20.4.20972905172...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{22}\cdot 41^{10}$
Root discriminant $65.48$
Ramified primes $2, 5, 41$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25759546399, 398435200, -20515534595, -80130810, 7517568315, -2598998, -1681061880, 2101610, 254889125, -247890, -27381447, 13110, 2107275, -230, -114580, -4, 4215, 0, -95, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 95*x^18 + 4215*x^16 - 4*x^15 - 114580*x^14 - 230*x^13 + 2107275*x^12 + 13110*x^11 - 27381447*x^10 - 247890*x^9 + 254889125*x^8 + 2101610*x^7 - 1681061880*x^6 - 2598998*x^5 + 7517568315*x^4 - 80130810*x^3 - 20515534595*x^2 + 398435200*x + 25759546399)
 
gp: K = bnfinit(x^20 - 95*x^18 + 4215*x^16 - 4*x^15 - 114580*x^14 - 230*x^13 + 2107275*x^12 + 13110*x^11 - 27381447*x^10 - 247890*x^9 + 254889125*x^8 + 2101610*x^7 - 1681061880*x^6 - 2598998*x^5 + 7517568315*x^4 - 80130810*x^3 - 20515534595*x^2 + 398435200*x + 25759546399, 1)
 

Normalized defining polynomial

\( x^{20} - 95 x^{18} + 4215 x^{16} - 4 x^{15} - 114580 x^{14} - 230 x^{13} + 2107275 x^{12} + 13110 x^{11} - 27381447 x^{10} - 247890 x^{9} + 254889125 x^{8} + 2101610 x^{7} - 1681061880 x^{6} - 2598998 x^{5} + 7517568315 x^{4} - 80130810 x^{3} - 20515534595 x^{2} + 398435200 x + 25759546399 \)

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:  \(2097290517211312656250000000000000000=2^{16}\cdot 5^{22}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.48$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} - \frac{1}{14} a^{16} + \frac{3}{14} a^{13} + \frac{1}{14} a^{12} + \frac{1}{14} a^{11} - \frac{1}{2} a^{9} + \frac{5}{14} a^{8} - \frac{1}{2} a^{7} - \frac{1}{14} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{5}{14} a^{3} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{300216513372847358194582469701626441721976618111129346738508496750049486} a^{19} + \frac{4727653650086257910727633058533326439335015826082597347688293953298101}{150108256686423679097291234850813220860988309055564673369254248375024743} a^{18} - \frac{29231790286535698224353537120240780413704361096972352385501315687004958}{150108256686423679097291234850813220860988309055564673369254248375024743} a^{17} - \frac{57514438831106807961540773503598719778351378144805856050975176451366007}{300216513372847358194582469701626441721976618111129346738508496750049486} a^{16} + \frac{2676974225784759669024831293233727996406203515383793531106613424586123}{42888073338978194027797495671660920245996659730161335248358356678578498} a^{15} + \frac{31114127076887870913201368177590234210185076891231935431909624238381632}{150108256686423679097291234850813220860988309055564673369254248375024743} a^{14} - \frac{11270799494134572449006329631770442846456573432410572718667236085965907}{100072171124282452731527489900542147240658872703709782246169498916683162} a^{13} + \frac{10035270481334098313423617838446050188984324898992517381249967668259610}{150108256686423679097291234850813220860988309055564673369254248375024743} a^{12} + \frac{10215386867509645366589758258535277307459440992036177216080982925292925}{150108256686423679097291234850813220860988309055564673369254248375024743} a^{11} - \frac{3506485091387944189544645839586680068155706176311479739415477484978959}{42888073338978194027797495671660920245996659730161335248358356678578498} a^{10} + \frac{23113766929037975250473025559635746766614660362984660261008622637542544}{150108256686423679097291234850813220860988309055564673369254248375024743} a^{9} - \frac{134099222181133632870504647276972803887453477857952067841778872718130217}{300216513372847358194582469701626441721976618111129346738508496750049486} a^{8} - \frac{9849455691043297603874545942130811173261543824235006852258486698502701}{33357390374760817577175829966847382413552957567903260748723166305561054} a^{7} - \frac{70353326423383658707223932803568702919707332279039114931594351856279977}{300216513372847358194582469701626441721976618111129346738508496750049486} a^{6} - \frac{21649286922298373954882173880077572725081890545863540737474517422454345}{150108256686423679097291234850813220860988309055564673369254248375024743} a^{5} - \frac{1448392970680578093596658236094605893000293015438927029736607422845}{877826062493705725715153420180194274040867304418506861808504376462133} a^{4} - \frac{15149875241839517834981226737911957773862852819148868219668372735112269}{100072171124282452731527489900542147240658872703709782246169498916683162} a^{3} + \frac{16260798291778614724813774051843709859807590221386050869568635740841403}{33357390374760817577175829966847382413552957567903260748723166305561054} a^{2} + \frac{141358503990615651605959784862272924251934919344803913242086181859161819}{300216513372847358194582469701626441721976618111129346738508496750049486} a + \frac{142105757506919875364127067496680063783099828859188398804169935061562297}{300216513372847358194582469701626441721976618111129346738508496750049486}$

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:  $11$
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:  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

$C_2\times F_5$ (as 20T13):

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

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{205}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{41})\), 5.1.50000.1, 10.2.289640502500000000.1, 10.2.1448202512500000000.1, 10.2.12500000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$41$41.10.5.1$x^{10} - 3362 x^{6} + 2825761 x^{2} - 5676953849$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
41.10.5.1$x^{10} - 3362 x^{6} + 2825761 x^{2} - 5676953849$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$