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Properties

Label 20.12.7833703334...7728.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{24}\cdot 13^{14}\cdot 17^{9}$
Root discriminant $49.51$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, 320, 160, -4208, -3872, 10576, 12560, -1072, -3728, -1864, -2052, -2624, 572, 2116, 476, -604, -180, 100, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 2*x^18 + 100*x^17 - 180*x^16 - 604*x^15 + 476*x^14 + 2116*x^13 + 572*x^12 - 2624*x^11 - 2052*x^10 - 1864*x^9 - 3728*x^8 - 1072*x^7 + 12560*x^6 + 10576*x^5 - 3872*x^4 - 4208*x^3 + 160*x^2 + 320*x - 16)
 
gp: K = bnfinit(x^20 - 4*x^19 + 2*x^18 + 100*x^17 - 180*x^16 - 604*x^15 + 476*x^14 + 2116*x^13 + 572*x^12 - 2624*x^11 - 2052*x^10 - 1864*x^9 - 3728*x^8 - 1072*x^7 + 12560*x^6 + 10576*x^5 - 3872*x^4 - 4208*x^3 + 160*x^2 + 320*x - 16, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 2 x^{18} + 100 x^{17} - 180 x^{16} - 604 x^{15} + 476 x^{14} + 2116 x^{13} + 572 x^{12} - 2624 x^{11} - 2052 x^{10} - 1864 x^{9} - 3728 x^{8} - 1072 x^{7} + 12560 x^{6} + 10576 x^{5} - 3872 x^{4} - 4208 x^{3} + 160 x^{2} + 320 x - 16 \)

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:  \(7833703334849737480841063435337728=2^{24}\cdot 13^{14}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.51$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{15}$, $\frac{1}{8} a^{16}$, $\frac{1}{8} a^{17}$, $\frac{1}{104} a^{18} - \frac{1}{52} a^{16} + \frac{1}{104} a^{15} + \frac{1}{104} a^{14} - \frac{3}{52} a^{13} + \frac{3}{52} a^{12} + \frac{3}{52} a^{11} - \frac{11}{52} a^{9} - \frac{1}{13} a^{8} + \frac{3}{26} a^{7} - \frac{1}{13} a^{6} - \frac{1}{13} a^{5} + \frac{7}{26} a^{4} + \frac{1}{13} a^{3} + \frac{3}{13} a + \frac{6}{13}$, $\frac{1}{32571744599087950596178170130864} a^{19} - \frac{24148236805405712064389315561}{16285872299543975298089085065432} a^{18} - \frac{148433102042587019621847843681}{8142936149771987649044542532716} a^{17} + \frac{300904051172537097986701334381}{8142936149771987649044542532716} a^{16} - \frac{765919360846113024649626050995}{16285872299543975298089085065432} a^{15} - \frac{6244739052270156694458568081}{16285872299543975298089085065432} a^{14} - \frac{250195720086081455986307190753}{8142936149771987649044542532716} a^{13} - \frac{722562599390674722891772418779}{8142936149771987649044542532716} a^{12} + \frac{90218904342044100799486939845}{2035734037442996912261135633179} a^{11} - \frac{63018069515385837626977219639}{8142936149771987649044542532716} a^{10} - \frac{1602603389683076649686971378587}{8142936149771987649044542532716} a^{9} - \frac{740536790721903336323273843065}{4071468074885993824522271266358} a^{8} - \frac{554273318600286911266986873783}{4071468074885993824522271266358} a^{7} + \frac{73109305607465740023114455331}{4071468074885993824522271266358} a^{6} + \frac{927984829408640503552291668799}{4071468074885993824522271266358} a^{5} + \frac{95850172641387358227394275201}{2035734037442996912261135633179} a^{4} + \frac{857889149034582973796583499744}{2035734037442996912261135633179} a^{3} - \frac{621770060887258885885849165865}{2035734037442996912261135633179} a^{2} + \frac{921656682938492308822601936410}{2035734037442996912261135633179} a - \frac{285381168395323447045930464043}{2035734037442996912261135633179}$

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:  \( 8590364849.08 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.10158928.1, 10.10.1341649635419392.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
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}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.5.1$x^{6} - 17$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$