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Properties

Label 20.12.1882282076...3328.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{42}\cdot 97^{5}\cdot 2657^{4}$
Root discriminant $65.12$
Ramified primes $2, 97, 2657$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T658

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-385724, 845536, 11528688, 18959776, -8943656, -34973776, -7190628, 20741056, 8443210, -5559644, -2987314, 757984, 518589, -54664, -50192, 1984, 2776, -28, -82, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 82*x^18 - 28*x^17 + 2776*x^16 + 1984*x^15 - 50192*x^14 - 54664*x^13 + 518589*x^12 + 757984*x^11 - 2987314*x^10 - 5559644*x^9 + 8443210*x^8 + 20741056*x^7 - 7190628*x^6 - 34973776*x^5 - 8943656*x^4 + 18959776*x^3 + 11528688*x^2 + 845536*x - 385724)
 
gp: K = bnfinit(x^20 - 82*x^18 - 28*x^17 + 2776*x^16 + 1984*x^15 - 50192*x^14 - 54664*x^13 + 518589*x^12 + 757984*x^11 - 2987314*x^10 - 5559644*x^9 + 8443210*x^8 + 20741056*x^7 - 7190628*x^6 - 34973776*x^5 - 8943656*x^4 + 18959776*x^3 + 11528688*x^2 + 845536*x - 385724, 1)
 

Normalized defining polynomial

\( x^{20} - 82 x^{18} - 28 x^{17} + 2776 x^{16} + 1984 x^{15} - 50192 x^{14} - 54664 x^{13} + 518589 x^{12} + 757984 x^{11} - 2987314 x^{10} - 5559644 x^{9} + 8443210 x^{8} + 20741056 x^{7} - 7190628 x^{6} - 34973776 x^{5} - 8943656 x^{4} + 18959776 x^{3} + 11528688 x^{2} + 845536 x - 385724 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1882282076712230748707114352092643328=2^{42}\cdot 97^{5}\cdot 2657^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.12$
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:  $2, 97, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{13} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{496} a^{18} - \frac{1}{248} a^{17} + \frac{29}{496} a^{16} - \frac{13}{124} a^{15} + \frac{3}{16} a^{14} + \frac{39}{248} a^{13} - \frac{97}{496} a^{12} + \frac{1}{124} a^{11} - \frac{3}{124} a^{10} + \frac{11}{31} a^{9} + \frac{35}{248} a^{8} + \frac{7}{31} a^{7} - \frac{1}{2} a^{6} - \frac{6}{31} a^{5} + \frac{19}{124} a^{4} - \frac{5}{31} a^{3} - \frac{43}{124} a^{2} - \frac{13}{62} a - \frac{45}{124}$, $\frac{1}{70115408370964721049356941044713994818863732595085026260112} a^{19} + \frac{34105053376264961581706684583348253337519349780524750171}{35057704185482360524678470522356997409431866297542513130056} a^{18} + \frac{3933529665435502245242470986775546526936306070814240811679}{70115408370964721049356941044713994818863732595085026260112} a^{17} + \frac{1634192987020590632998706659923930801045936809569289793785}{35057704185482360524678470522356997409431866297542513130056} a^{16} + \frac{1927416651267877438642946489776809730400086105840792245641}{70115408370964721049356941044713994818863732595085026260112} a^{15} - \frac{3528373079306695000597198863787630184038420227349670286651}{35057704185482360524678470522356997409431866297542513130056} a^{14} - \frac{3690355633191020122149936862594334204370886308347311888931}{70115408370964721049356941044713994818863732595085026260112} a^{13} + \frac{4766280141114411459528248088168787785560597041552348808511}{35057704185482360524678470522356997409431866297542513130056} a^{12} + \frac{10495016491988416639678386314140527858397431936251117629}{82683264588401793690279411609332541059980816739487059269} a^{11} + \frac{1588582148105082520286792285813255147234238552896003949}{221884203705584560282775129888335426641973837326218437532} a^{10} + \frac{2708555855510790554938507686095921724558874461631684349437}{35057704185482360524678470522356997409431866297542513130056} a^{9} + \frac{4576176834588804091793290597289965510805719273199873102251}{17528852092741180262339235261178498704715933148771256565028} a^{8} + \frac{69734504866544504132483465303946422651986273462303279929}{8764426046370590131169617630589249352357966574385628282514} a^{7} + \frac{1152579905759203212545207658617087465356992483960810535797}{8764426046370590131169617630589249352357966574385628282514} a^{6} + \frac{4049136823656949424069386695312388651650986093031838684421}{17528852092741180262339235261178498704715933148771256565028} a^{5} - \frac{1144415688688458032105212449263485937091970215519219116139}{8764426046370590131169617630589249352357966574385628282514} a^{4} - \frac{5060307079125478655615820235542311443873714881919282525335}{17528852092741180262339235261178498704715933148771256565028} a^{3} + \frac{3156347672360561028085044123986733914266126798344650076293}{8764426046370590131169617630589249352357966574385628282514} a^{2} + \frac{4610412195734847093154458492573373481602224811803879416661}{17528852092741180262339235261178498704715933148771256565028} a - \frac{844787162117282647914581276609459655972750520601527409025}{8764426046370590131169617630589249352357966574385628282514}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 176154275023 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T658:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 76 conjugacy class representatives for t20n658 are not computed
Character table for t20n658 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.24832.1, 10.6.925322313728.1

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 24 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.12.26.64$x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{6} - 2 x^{4} + 4 x^{3} + 2$$12$$1$$26$$S_3 \times C_2^2$$[2, 3]_{3}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
2657Data not computed