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Properties

Label 20.4.15241120055...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{38}\cdot 5^{12}\cdot 17^{6}\cdot 97^{2}$
Root discriminant $36.24$
Ramified primes $2, 5, 17, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 544, 1924, 5120, 7760, 4472, -4330, -5268, 3596, 4248, -4619, -962, 2419, -952, 142, -236, 172, -24, 7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 7*x^18 - 24*x^17 + 172*x^16 - 236*x^15 + 142*x^14 - 952*x^13 + 2419*x^12 - 962*x^11 - 4619*x^10 + 4248*x^9 + 3596*x^8 - 5268*x^7 - 4330*x^6 + 4472*x^5 + 7760*x^4 + 5120*x^3 + 1924*x^2 + 544*x + 64)
 
gp: K = bnfinit(x^20 - 6*x^19 + 7*x^18 - 24*x^17 + 172*x^16 - 236*x^15 + 142*x^14 - 952*x^13 + 2419*x^12 - 962*x^11 - 4619*x^10 + 4248*x^9 + 3596*x^8 - 5268*x^7 - 4330*x^6 + 4472*x^5 + 7760*x^4 + 5120*x^3 + 1924*x^2 + 544*x + 64, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 7 x^{18} - 24 x^{17} + 172 x^{16} - 236 x^{15} + 142 x^{14} - 952 x^{13} + 2419 x^{12} - 962 x^{11} - 4619 x^{10} + 4248 x^{9} + 3596 x^{8} - 5268 x^{7} - 4330 x^{6} + 4472 x^{5} + 7760 x^{4} + 5120 x^{3} + 1924 x^{2} + 544 x + 64 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15241120055446994944000000000000=2^{38}\cdot 5^{12}\cdot 17^{6}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.24$
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:  $2, 5, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{20} a^{16} - \frac{1}{5} a^{14} - \frac{1}{10} a^{13} - \frac{1}{10} a^{12} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{20} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{20} a^{17} - \frac{1}{5} a^{15} - \frac{1}{10} a^{14} - \frac{1}{10} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{20} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{20} a^{18} - \frac{1}{10} a^{15} + \frac{1}{10} a^{14} + \frac{1}{10} a^{13} - \frac{1}{10} a^{12} + \frac{1}{5} a^{11} + \frac{3}{20} a^{10} - \frac{3}{10} a^{7} + \frac{3}{10} a^{6} + \frac{3}{10} a^{5} - \frac{1}{10} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{9515697908469960604355701792053760} a^{19} - \frac{17921403541238020530317942171809}{951569790846996060435570179205376} a^{18} + \frac{148741122300439566756623233086127}{9515697908469960604355701792053760} a^{17} - \frac{288268547836621145283427985943}{67969270774785432888255012800384} a^{16} - \frac{12086643113434040999591199801729}{2378924477117490151088925448013440} a^{15} - \frac{49891179766108377871018601307275}{475784895423498030217785089602688} a^{14} + \frac{176942558411307700822361251823667}{951569790846996060435570179205376} a^{13} - \frac{27219775053602496214995438802807}{118946223855874507554446272400672} a^{12} + \frac{326751558681377898785781910474455}{1903139581693992120871140358410752} a^{11} + \frac{32186730917689751606871360882757}{951569790846996060435570179205376} a^{10} + \frac{1982103199543009320345467819437741}{9515697908469960604355701792053760} a^{9} + \frac{49654097466812772932627355996309}{475784895423498030217785089602688} a^{8} - \frac{707347112859875795806016698611937}{2378924477117490151088925448013440} a^{7} - \frac{172810189282621978034325281693677}{475784895423498030217785089602688} a^{6} + \frac{158801585208434254332859779533587}{4757848954234980302177850896026880} a^{5} + \frac{89993803750486130219349383165411}{297365559639686268886115681001680} a^{4} + \frac{1016373707512550526931710220201}{118946223855874507554446272400672} a^{3} + \frac{5661684039153900671284988448771}{148682779819843134443057840500840} a^{2} + \frac{5830019982103463656426225636203}{67969270774785432888255012800384} a + \frac{202299782093679011017360936656433}{594731119279372537772231362003360}$

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:  $11$
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:  \( 410568239.888 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.2.2367488000000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.6.4$x^{8} + 136 x^{4} + 7803$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
97Data not computed