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Properties

Label 20.0.14044288593...6624.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 881^{8}$
Root discriminant $45.43$
Ramified primes $2, 3, 881$
Class number $36$ (GRH)
Class group $[6, 6]$ (GRH)
Galois group 20T277

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![104976, 0, -233280, 0, 351216, 0, -269784, 0, 146412, 0, -51096, 0, 13093, 0, -2265, 0, 284, 0, -21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 21*x^18 + 284*x^16 - 2265*x^14 + 13093*x^12 - 51096*x^10 + 146412*x^8 - 269784*x^6 + 351216*x^4 - 233280*x^2 + 104976)
 
gp: K = bnfinit(x^20 - 21*x^18 + 284*x^16 - 2265*x^14 + 13093*x^12 - 51096*x^10 + 146412*x^8 - 269784*x^6 + 351216*x^4 - 233280*x^2 + 104976, 1)
 

Normalized defining polynomial

\( x^{20} - 21 x^{18} + 284 x^{16} - 2265 x^{14} + 13093 x^{12} - 51096 x^{10} + 146412 x^{8} - 269784 x^{6} + 351216 x^{4} - 233280 x^{2} + 104976 \)

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:  \(1404428859331790362883274100506624=2^{16}\cdot 3^{10}\cdot 881^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.43$
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:  $2, 3, 881$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{12} a^{11} + \frac{1}{6} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{36} a^{12} - \frac{1}{9} a^{8} - \frac{1}{4} a^{6} + \frac{4}{9} a^{4}$, $\frac{1}{108} a^{13} + \frac{1}{36} a^{11} + \frac{7}{54} a^{9} + \frac{5}{36} a^{7} - \frac{29}{108} a^{5} - \frac{1}{18} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{216} a^{14} - \frac{1}{216} a^{13} + \frac{1}{36} a^{11} - \frac{1}{54} a^{10} + \frac{5}{27} a^{9} + \frac{1}{8} a^{8} - \frac{17}{72} a^{7} + \frac{2}{27} a^{6} - \frac{53}{108} a^{5} - \frac{1}{18} a^{3} + \frac{1}{6} a$, $\frac{1}{648} a^{15} - \frac{1}{216} a^{13} - \frac{1}{162} a^{11} + \frac{13}{216} a^{9} + \frac{151}{648} a^{7} - \frac{2}{27} a^{5} - \frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{648} a^{16} - \frac{1}{216} a^{13} - \frac{1}{162} a^{12} + \frac{1}{36} a^{11} + \frac{1}{24} a^{10} + \frac{5}{27} a^{9} - \frac{23}{162} a^{8} - \frac{17}{72} a^{7} + \frac{1}{108} a^{5} - \frac{1}{6} a^{4} - \frac{1}{18} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a$, $\frac{1}{3888} a^{17} - \frac{1}{1296} a^{15} - \frac{1}{972} a^{13} - \frac{1}{72} a^{12} + \frac{13}{1296} a^{11} + \frac{799}{3888} a^{9} + \frac{1}{18} a^{8} - \frac{1}{81} a^{7} + \frac{1}{8} a^{6} + \frac{11}{36} a^{5} + \frac{5}{18} a^{4} - \frac{13}{36} a^{3} - \frac{1}{2}$, $\frac{1}{3698605676451024} a^{18} - \frac{713738165881}{1232868558817008} a^{16} + \frac{2466343759489}{1849302838225512} a^{14} - \frac{9192821837519}{1232868558817008} a^{12} - \frac{197742014872985}{3698605676451024} a^{10} + \frac{112787817877183}{616434279408504} a^{8} - \frac{6102428117191}{25684761642021} a^{6} - \frac{3907153630529}{34246348856028} a^{4} - \frac{1}{2} a^{3} + \frac{1753539750647}{5707724809338} a^{2} + \frac{4913693083}{317095822741}$, $\frac{1}{11095817029353072} a^{19} + \frac{118774651171}{1849302838225512} a^{17} - \frac{3628899695029}{11095817029353072} a^{15} - \frac{12997971710411}{3698605676451024} a^{13} - \frac{1}{72} a^{12} + \frac{110887879306679}{5547908514676536} a^{11} + \frac{369220043456039}{3698605676451024} a^{9} + \frac{1}{18} a^{8} - \frac{15615302799421}{77054284926063} a^{7} + \frac{1}{8} a^{6} - \frac{15322603249205}{102739046568084} a^{5} - \frac{2}{9} a^{4} + \frac{12068666715301}{34246348856028} a^{3} - \frac{1}{2} a^{2} - \frac{104060709886}{317095822741} a - \frac{1}{2}$

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

Class group and class number

$C_{6}\times C_{6}$, which has order $36$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{35535368219}{3698605676451024} a^{18} + \frac{238152122741}{1232868558817008} a^{16} - \frac{592628158306}{231162854778189} a^{14} + \frac{24181071499045}{1232868558817008} a^{12} - \frac{407372281669979}{3698605676451024} a^{10} + \frac{125093180384027}{308217139704252} a^{8} - \frac{28730262266062}{25684761642021} a^{6} + \frac{63192864475531}{34246348856028} a^{4} - \frac{7287553300949}{2853862404669} a^{2} + \frac{532762285705}{317095822741} \) (order $6$)
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:  \( 43831645.0272 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T277:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 48 conjugacy class representatives for t20n277
Character table for t20n277 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.3104644.1, 10.10.4163967806429952.1, 10.0.2342231891116848.1, 10.0.1387989268809984.2

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.12.16.19$x^{12} + x^{10} - 2 x^{8} - 3 x^{6} + 2 x^{4} - 3 x^{2} + 1$$6$$2$$16$$C_2\times S_4$$[4/3, 4/3, 2]_{3}^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
881Data not computed