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Properties

Label 20.0.82186144118...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{22}\cdot 47^{10}$
Root discriminant $70.10$
Ramified primes $2, 5, 47$
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![26151329351, -361510800, 26550173765, -145608970, 12067309335, -5313886, 3169335980, 3402250, 529079305, 468870, 58615057, 21030, 4369535, 210, 216740, -4, 6855, 0, 125, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 125*x^18 + 6855*x^16 - 4*x^15 + 216740*x^14 + 210*x^13 + 4369535*x^12 + 21030*x^11 + 58615057*x^10 + 468870*x^9 + 529079305*x^8 + 3402250*x^7 + 3169335980*x^6 - 5313886*x^5 + 12067309335*x^4 - 145608970*x^3 + 26550173765*x^2 - 361510800*x + 26151329351)
 
gp: K = bnfinit(x^20 + 125*x^18 + 6855*x^16 - 4*x^15 + 216740*x^14 + 210*x^13 + 4369535*x^12 + 21030*x^11 + 58615057*x^10 + 468870*x^9 + 529079305*x^8 + 3402250*x^7 + 3169335980*x^6 - 5313886*x^5 + 12067309335*x^4 - 145608970*x^3 + 26550173765*x^2 - 361510800*x + 26151329351, 1)
 

Normalized defining polynomial

\( x^{20} + 125 x^{18} + 6855 x^{16} - 4 x^{15} + 216740 x^{14} + 210 x^{13} + 4369535 x^{12} + 21030 x^{11} + 58615057 x^{10} + 468870 x^{9} + 529079305 x^{8} + 3402250 x^{7} + 3169335980 x^{6} - 5313886 x^{5} + 12067309335 x^{4} - 145608970 x^{3} + 26550173765 x^{2} - 361510800 x + 26151329351 \)

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:  \(8218614411848445156250000000000000000=2^{16}\cdot 5^{22}\cdot 47^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.10$
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:  $2, 5, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} + \frac{1}{14} a^{16} - \frac{3}{14} a^{15} - \frac{3}{14} a^{14} - \frac{1}{7} a^{13} - \frac{1}{7} a^{12} - \frac{1}{14} a^{11} - \frac{1}{7} a^{10} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{2} a^{7} - \frac{1}{14} a^{6} + \frac{1}{14} a^{5} + \frac{3}{14} a^{4} - \frac{3}{7} a^{3} + \frac{1}{14} a^{2} - \frac{3}{7} a$, $\frac{1}{4620781349327861035510336137568281612229590834425329195225987630828092802} a^{19} - \frac{135425837146644977040011385261620183015891878032086998026925222022586137}{4620781349327861035510336137568281612229590834425329195225987630828092802} a^{18} + \frac{1125688615712597739427974208743679335273703332975946786285936300700477211}{4620781349327861035510336137568281612229590834425329195225987630828092802} a^{17} - \frac{410235235545468643055884242743654121504328076837073951868148678092328402}{2310390674663930517755168068784140806114795417212664597612993815414046401} a^{16} - \frac{403904009341559581825692155379933690585351000982610946840956020288767230}{2310390674663930517755168068784140806114795417212664597612993815414046401} a^{15} + \frac{158293472508860485811910045947424383787957596675265601174690162763803929}{2310390674663930517755168068784140806114795417212664597612993815414046401} a^{14} - \frac{333831899549976383447017195278571329306054675746834665863066948533573511}{2310390674663930517755168068784140806114795417212664597612993815414046401} a^{13} - \frac{352866224621481644542679838646991250483096938606479542837439635966430649}{2310390674663930517755168068784140806114795417212664597612993815414046401} a^{12} + \frac{44223952461110596481831409856672726247155340533552444343404064423903953}{4620781349327861035510336137568281612229590834425329195225987630828092802} a^{11} + \frac{389839527897843942301275884745175755589059567781411166324091678726096750}{2310390674663930517755168068784140806114795417212664597612993815414046401} a^{10} - \frac{1008104938307466608394906294786625284507424181814211672715755541614323131}{2310390674663930517755168068784140806114795417212664597612993815414046401} a^{9} + \frac{26104012777880837896417054102499586253169836217576917628735111819451047}{4620781349327861035510336137568281612229590834425329195225987630828092802} a^{8} + \frac{219597314120092594556564539490695972319986337448220262866429982573348785}{2310390674663930517755168068784140806114795417212664597612993815414046401} a^{7} - \frac{208184249691111377610964787326407317273085688243703836305448335315453375}{660111621332551576501476591081183087461370119203618456460855375832584686} a^{6} + \frac{49244321479317961088057937576645398144647416085888734858121328810214837}{200903536927298305891753745111664417923025688453275182401129896992525774} a^{5} - \frac{1624613485442020850310380572518583075023929192027491173671439040538970685}{4620781349327861035510336137568281612229590834425329195225987630828092802} a^{4} - \frac{2223663801892652863771584726378587889300515175975292899773737750100517139}{4620781349327861035510336137568281612229590834425329195225987630828092802} a^{3} + \frac{229448451272219541305232310532984206301050763019158458339380563587784520}{2310390674663930517755168068784140806114795417212664597612993815414046401} a^{2} + \frac{9771168318762742858242158488376646059798049741232601147675938172703878}{100451768463649152945876872555832208961512844226637591200564948496262887} a - \frac{178182827568134127476646201734735318235279885368573836843122155303779119}{660111621332551576501476591081183087461370119203618456460855375832584686}$

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

$C_2\times F_5$ (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{-235}) \), \(\Q(\sqrt{5}, \sqrt{-47})\), 5.1.50000.1, 10.2.12500000000.1, 10.0.573362517500000000.1, 10.0.2866812587500000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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}$
5Data not computed
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$