Properties

Label 20.0.36235542197...7561.1
Degree $20$
Signature $[0, 10]$
Discriminant $47^{10}\cdot 83^{2}$
Root discriminant $10.66$
Ramified primes $47, 83$
Class number $1$
Class group Trivial
Galois Group $C_2\times C_2^4:D_5$ (as 20T81)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 4, 1, -8, 8, 17, -13, 3, 3, -3, 8, -1, 8, -7, 5, -4, 3, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 - 4*x^17 + 5*x^16 - 7*x^15 + 8*x^14 - x^13 + 8*x^12 - 3*x^11 + 3*x^10 + 3*x^9 - 13*x^8 + 17*x^7 + 8*x^6 - 8*x^5 + x^4 + 4*x^3 + 1)
gp: K = bnfinit(x^20 - x^19 + 3*x^18 - 4*x^17 + 5*x^16 - 7*x^15 + 8*x^14 - x^13 + 8*x^12 - 3*x^11 + 3*x^10 + 3*x^9 - 13*x^8 + 17*x^7 + 8*x^6 - 8*x^5 + x^4 + 4*x^3 + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut x^{19} \) \(\mathstrut +\mathstrut 3 x^{18} \) \(\mathstrut -\mathstrut 4 x^{17} \) \(\mathstrut +\mathstrut 5 x^{16} \) \(\mathstrut -\mathstrut 7 x^{15} \) \(\mathstrut +\mathstrut 8 x^{14} \) \(\mathstrut -\mathstrut x^{13} \) \(\mathstrut +\mathstrut 8 x^{12} \) \(\mathstrut -\mathstrut 3 x^{11} \) \(\mathstrut +\mathstrut 3 x^{10} \) \(\mathstrut +\mathstrut 3 x^{9} \) \(\mathstrut -\mathstrut 13 x^{8} \) \(\mathstrut +\mathstrut 17 x^{7} \) \(\mathstrut +\mathstrut 8 x^{6} \) \(\mathstrut -\mathstrut 8 x^{5} \) \(\mathstrut +\mathstrut x^{4} \) \(\mathstrut +\mathstrut 4 x^{3} \) \(\mathstrut +\mathstrut 1 \)

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:  \(362355421972633207561=47^{10}\cdot 83^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.66$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $47, 83$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{17} - \frac{1}{5} a^{16} - \frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{2}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{2194608515} a^{19} + \frac{74558294}{2194608515} a^{18} - \frac{1017294346}{2194608515} a^{17} + \frac{6939284}{39901973} a^{16} + \frac{856912424}{2194608515} a^{15} - \frac{994013257}{2194608515} a^{14} - \frac{586393183}{2194608515} a^{13} - \frac{382352308}{2194608515} a^{12} + \frac{9694604}{438921703} a^{11} - \frac{654463201}{2194608515} a^{10} + \frac{509560233}{2194608515} a^{9} + \frac{127710497}{2194608515} a^{8} - \frac{19417716}{438921703} a^{7} + \frac{941830149}{2194608515} a^{6} - \frac{675824012}{2194608515} a^{5} - \frac{811533444}{2194608515} a^{4} - \frac{1050222631}{2194608515} a^{3} + \frac{195415228}{438921703} a^{2} + \frac{339804044}{2194608515} a - \frac{29307092}{438921703}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{17167678}{39901973} a^{19} - \frac{198448696}{199509865} a^{18} + \frac{459542381}{199509865} a^{17} - \frac{840311969}{199509865} a^{16} + \frac{260122109}{39901973} a^{15} - \frac{1849657884}{199509865} a^{14} + \frac{2532135297}{199509865} a^{13} - \frac{2478167597}{199509865} a^{12} + \frac{2760348783}{199509865} a^{11} - \frac{578017673}{39901973} a^{10} + \frac{2728324706}{199509865} a^{9} - \frac{1951319303}{199509865} a^{8} + \frac{521684558}{199509865} a^{7} + \frac{368598185}{39901973} a^{6} - \frac{1381145139}{199509865} a^{5} + \frac{302967502}{199509865} a^{4} + \frac{53721484}{199509865} a^{3} + \frac{280222406}{199509865} a^{2} - \frac{56123045}{39901973} a + \frac{294207701}{199509865} \),  \( \frac{35747331}{438921703} a^{19} - \frac{145326186}{2194608515} a^{18} + \frac{403294346}{2194608515} a^{17} - \frac{16979709}{199509865} a^{16} + \frac{12943137}{438921703} a^{15} + \frac{338271231}{2194608515} a^{14} - \frac{1035587648}{2194608515} a^{13} + \frac{2580202278}{2194608515} a^{12} - \frac{2523174922}{2194608515} a^{11} + \frac{644125245}{438921703} a^{10} - \frac{1462277224}{2194608515} a^{9} + \frac{4268504467}{2194608515} a^{8} - \frac{4522480687}{2194608515} a^{7} + \frac{416180518}{438921703} a^{6} + \frac{2431179411}{2194608515} a^{5} - \frac{9009900908}{2194608515} a^{4} + \frac{6052739344}{2194608515} a^{3} + \frac{2462644731}{2194608515} a^{2} - \frac{267045956}{438921703} a - \frac{225933434}{2194608515} \),  \( \frac{140164190}{438921703} a^{19} - \frac{3029118987}{2194608515} a^{18} + \frac{5495443702}{2194608515} a^{17} - \frac{1056094673}{199509865} a^{16} + \frac{3543454491}{438921703} a^{15} - \frac{24383649848}{2194608515} a^{14} + \frac{34129248594}{2194608515} a^{13} - \frac{36148677594}{2194608515} a^{12} + \frac{30310058261}{2194608515} a^{11} - \frac{7978649180}{438921703} a^{10} + \frac{33598390482}{2194608515} a^{9} - \frac{25547226321}{2194608515} a^{8} + \frac{7023841916}{2194608515} a^{7} + \frac{6150886199}{438921703} a^{6} - \frac{34791286373}{2194608515} a^{5} - \frac{2801703951}{2194608515} a^{4} + \frac{9638984958}{2194608515} a^{3} + \frac{2035043012}{2194608515} a^{2} - \frac{595370537}{438921703} a + \frac{2421876982}{2194608515} \),  \( \frac{599914055}{438921703} a^{19} - \frac{6328764117}{2194608515} a^{18} + \frac{14781313397}{2194608515} a^{17} - \frac{2359285278}{199509865} a^{16} + \frac{7763200751}{438921703} a^{15} - \frac{54448405703}{2194608515} a^{14} + \frac{72218992069}{2194608515} a^{13} - \frac{65019050674}{2194608515} a^{12} + \frac{72818839881}{2194608515} a^{11} - \frac{14485773913}{438921703} a^{10} + \frac{68365953842}{2194608515} a^{9} - \frac{42559432106}{2194608515} a^{8} - \frac{5470702269}{2194608515} a^{7} + \frac{14242725327}{438921703} a^{6} - \frac{43343911258}{2194608515} a^{5} - \frac{4552730366}{2194608515} a^{4} + \frac{15215114478}{2194608515} a^{3} + \frac{4852902172}{2194608515} a^{2} - \frac{1309620302}{438921703} a + \frac{5352449662}{2194608515} \),  \( \frac{5352449662}{2194608515} a^{19} - \frac{8352019937}{2194608515} a^{18} + \frac{22386113103}{2194608515} a^{17} - \frac{658020219}{39901973} a^{16} + \frac{52714386368}{2194608515} a^{15} - \frac{76283151389}{2194608515} a^{14} + \frac{97268002999}{2194608515} a^{13} - \frac{77571441731}{2194608515} a^{12} + \frac{21567729594}{438921703} a^{11} - \frac{88876188867}{2194608515} a^{10} + \frac{88486218551}{2194608515} a^{9} - \frac{52308604856}{2194608515} a^{8} - \frac{5404482700}{438921703} a^{7} + \frac{96462346523}{2194608515} a^{6} - \frac{28394029339}{2194608515} a^{5} + \frac{524313962}{2194608515} a^{4} + \frac{9905180028}{2194608515} a^{3} + \frac{1238936834}{438921703} a^{2} - \frac{4852902172}{2194608515} a + \frac{1309620302}{438921703} \),  \( \frac{2707494034}{2194608515} a^{19} - \frac{5555157859}{2194608515} a^{18} + \frac{12429096146}{2194608515} a^{17} - \frac{400515119}{39901973} a^{16} + \frac{31963004506}{2194608515} a^{15} - \frac{44760304288}{2194608515} a^{14} + \frac{59543687813}{2194608515} a^{13} - \frac{50880774287}{2194608515} a^{12} + \frac{11611246756}{438921703} a^{11} - \frac{61335888844}{2194608515} a^{10} + \frac{53948495212}{2194608515} a^{9} - \frac{33498872742}{2194608515} a^{8} - \frac{1695935381}{438921703} a^{7} + \frac{61586534701}{2194608515} a^{6} - \frac{26555629888}{2194608515} a^{5} - \frac{15417470496}{2194608515} a^{4} + \frac{8340201046}{2194608515} a^{3} + \frac{2034866506}{438921703} a^{2} - \frac{6407575579}{2194608515} a + \frac{507156570}{438921703} \),  \( \frac{2529411972}{2194608515} a^{19} - \frac{5532330713}{2194608515} a^{18} + \frac{12279972399}{2194608515} a^{17} - \frac{1995362534}{199509865} a^{16} + \frac{32003869343}{2194608515} a^{15} - \frac{44568041303}{2194608515} a^{14} + \frac{58830908611}{2194608515} a^{13} - \frac{51404782113}{2194608515} a^{12} + \frac{55323097553}{2194608515} a^{11} - \frac{57669217082}{2194608515} a^{10} + \frac{50985256052}{2194608515} a^{9} - \frac{30691585729}{2194608515} a^{8} - \frac{11694648927}{2194608515} a^{7} + \frac{65504322668}{2194608515} a^{6} - \frac{38687198478}{2194608515} a^{5} - \frac{11043805931}{2194608515} a^{4} + \frac{15212489607}{2194608515} a^{3} + \frac{4352691061}{2194608515} a^{2} - \frac{6269809532}{2194608515} a + \frac{3185400421}{2194608515} \),  \( \frac{3769389187}{2194608515} a^{19} - \frac{5479438171}{2194608515} a^{18} + \frac{14363617537}{2194608515} a^{17} - \frac{2064386926}{199509865} a^{16} + \frac{31905425633}{2194608515} a^{15} - \frac{9127974501}{438921703} a^{14} + \frac{57768211187}{2194608515} a^{13} - \frac{40322926184}{2194608515} a^{12} + \frac{61538631897}{2194608515} a^{11} - \frac{51031028432}{2194608515} a^{10} + \frac{9741648969}{438921703} a^{9} - \frac{23774998318}{2194608515} a^{8} - \frac{25925501378}{2194608515} a^{7} + \frac{65878753533}{2194608515} a^{6} - \frac{622046301}{438921703} a^{5} - \frac{3489893156}{438921703} a^{4} + \frac{5043566969}{2194608515} a^{3} + \frac{13483188839}{2194608515} a^{2} - \frac{4861363692}{2194608515} a + \frac{2302714179}{2194608515} \),  \( \frac{904504175}{438921703} a^{19} - \frac{7684902578}{2194608515} a^{18} + \frac{19536109453}{2194608515} a^{17} - \frac{2933794162}{199509865} a^{16} + \frac{9416070745}{438921703} a^{15} - \frac{66918843862}{2194608515} a^{14} + \frac{86368648006}{2194608515} a^{13} - \frac{69506968761}{2194608515} a^{12} + \frac{90524548749}{2194608515} a^{11} - \frac{15815417419}{438921703} a^{10} + \frac{75705925198}{2194608515} a^{9} - \frac{42994422364}{2194608515} a^{8} - \frac{23685749551}{2194608515} a^{7} + \frac{18733010240}{438921703} a^{6} - \frac{32577094552}{2194608515} a^{5} - \frac{4564214254}{2194608515} a^{4} + \frac{12290055387}{2194608515} a^{3} + \frac{3849438258}{2194608515} a^{2} - \frac{756814606}{438921703} a + \frac{5327087823}{2194608515} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 59.602445011 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\times C_2^4:D_5$ (as 20T81):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{-47}) \), 5.1.2209.1 x5, 10.0.405013523.1, 10.2.19035635581.1, 10.0.229345007.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$