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Properties

Label 20.0.21883072113...5424.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{15}\cdot 7^{5}\cdot 61^{4}$
Root discriminant $14.69$
Ramified primes $2, 3, 7, 61$
Class number $1$
Class group Trivial
Galois group 20T299

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 43, -119, 254, -469, 715, -825, 748, -625, 497, -321, 174, -115, 85, -47, 22, -15, 11, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 11*x^18 - 15*x^17 + 22*x^16 - 47*x^15 + 85*x^14 - 115*x^13 + 174*x^12 - 321*x^11 + 497*x^10 - 625*x^9 + 748*x^8 - 825*x^7 + 715*x^6 - 469*x^5 + 254*x^4 - 119*x^3 + 43*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 11*x^18 - 15*x^17 + 22*x^16 - 47*x^15 + 85*x^14 - 115*x^13 + 174*x^12 - 321*x^11 + 497*x^10 - 625*x^9 + 748*x^8 - 825*x^7 + 715*x^6 - 469*x^5 + 254*x^4 - 119*x^3 + 43*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 11 x^{18} - 15 x^{17} + 22 x^{16} - 47 x^{15} + 85 x^{14} - 115 x^{13} + 174 x^{12} - 321 x^{11} + 497 x^{10} - 625 x^{9} + 748 x^{8} - 825 x^{7} + 715 x^{6} - 469 x^{5} + 254 x^{4} - 119 x^{3} + 43 x^{2} - 9 x + 1 \)

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:  \(218830721135617121255424=2^{16}\cdot 3^{15}\cdot 7^{5}\cdot 61^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.69$
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:  $2, 3, 7, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{2}{5} a^{17} + \frac{1}{5} a^{16} - \frac{1}{5} a^{15} + \frac{1}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{22417709162965} a^{19} + \frac{1636585132639}{22417709162965} a^{18} - \frac{140728590857}{4483541832593} a^{17} + \frac{4103617986066}{22417709162965} a^{16} + \frac{1545415248084}{22417709162965} a^{15} - \frac{4489200483089}{22417709162965} a^{14} - \frac{3390288994153}{22417709162965} a^{13} - \frac{1476421396543}{4483541832593} a^{12} + \frac{10685217777326}{22417709162965} a^{11} + \frac{6077077860759}{22417709162965} a^{10} - \frac{4941807318201}{22417709162965} a^{9} + \frac{1514669724816}{22417709162965} a^{8} - \frac{3156432626476}{22417709162965} a^{7} + \frac{840176376693}{4483541832593} a^{6} - \frac{1344841959331}{22417709162965} a^{5} - \frac{7986229111371}{22417709162965} a^{4} - \frac{5704444972587}{22417709162965} a^{3} + \frac{8564675747304}{22417709162965} a^{2} + \frac{8651091954061}{22417709162965} a - \frac{9919584255397}{22417709162965}$

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

Class group and class number

Trivial group, which has 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:  \( \frac{57422048}{83100881} a^{19} - \frac{1312374107}{415504405} a^{18} + \frac{2630525921}{415504405} a^{17} - \frac{3301735622}{415504405} a^{16} + \frac{5097881172}{415504405} a^{15} - \frac{11535096047}{415504405} a^{14} + \frac{19900323817}{415504405} a^{13} - \frac{25498305273}{415504405} a^{12} + \frac{40499899656}{415504405} a^{11} - \frac{76609420649}{415504405} a^{10} + \frac{22661147457}{83100881} a^{9} - \frac{137109358323}{415504405} a^{8} + \frac{164169296934}{415504405} a^{7} - \frac{176297850441}{415504405} a^{6} + \frac{141686160017}{415504405} a^{5} - \frac{85788978232}{415504405} a^{4} + \frac{44698897896}{415504405} a^{3} - \frac{19897700873}{415504405} a^{2} + \frac{6569872123}{415504405} a - \frac{738992428}{415504405} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5970.91953124 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T299:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5000
The 230 conjugacy class representatives for t20n299 are not computed
Character table for t20n299 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.189.1

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 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 ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
3Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.5.4.3$x^{5} - 244$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.0.1$x^{5} - x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$