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Properties

Label 20.6.12543323349...0000.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{28}\cdot 3^{4}\cdot 5^{10}\cdot 17^{4}\cdot 29^{4}$
Root discriminant $25.41$
Ramified primes $2, 3, 5, 17, 29$
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![-1, 0, 2549, 0, -2397, 0, -2834, 0, 2271, 0, 307, 0, -559, 0, 94, 0, 29, 0, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 11*x^18 + 29*x^16 + 94*x^14 - 559*x^12 + 307*x^10 + 2271*x^8 - 2834*x^6 - 2397*x^4 + 2549*x^2 - 1)
 
gp: K = bnfinit(x^20 - 11*x^18 + 29*x^16 + 94*x^14 - 559*x^12 + 307*x^10 + 2271*x^8 - 2834*x^6 - 2397*x^4 + 2549*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 11 x^{18} + 29 x^{16} + 94 x^{14} - 559 x^{12} + 307 x^{10} + 2271 x^{8} - 2834 x^{6} - 2397 x^{4} + 2549 x^{2} - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-12543323349925232640000000000=-\,2^{28}\cdot 3^{4}\cdot 5^{10}\cdot 17^{4}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.41$
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:  $2, 3, 5, 17, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{8} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{7} - \frac{1}{8} a^{3} - \frac{1}{8} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} + \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{8} + \frac{1}{16} a^{7} + \frac{3}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} - \frac{1}{8} a^{2} - \frac{3}{16} a - \frac{5}{16}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{4} a^{7} + \frac{1}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{3}{16} a^{3} - \frac{3}{16} a^{2} - \frac{1}{2} a - \frac{3}{16}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{10} - \frac{1}{16} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} - \frac{1}{16}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} - \frac{1}{16} a$, $\frac{1}{667483424} a^{18} - \frac{1}{32} a^{17} + \frac{19486617}{667483424} a^{16} - \frac{2896343}{333741712} a^{14} - \frac{1}{16} a^{13} + \frac{20104671}{333741712} a^{12} - \frac{1}{8} a^{11} + \frac{7420519}{667483424} a^{10} - \frac{3}{32} a^{9} - \frac{45721425}{667483424} a^{8} + \frac{1}{8} a^{7} + \frac{17858287}{333741712} a^{6} - \frac{1}{16} a^{5} + \frac{55950455}{333741712} a^{4} - \frac{1}{8} a^{3} - \frac{113364723}{667483424} a^{2} - \frac{9}{32} a - \frac{220797903}{667483424}$, $\frac{1}{667483424} a^{19} - \frac{85765}{41717714} a^{17} - \frac{1}{32} a^{16} - \frac{2896343}{333741712} a^{15} - \frac{377093}{166870856} a^{13} - \frac{1}{16} a^{12} - \frac{76014909}{667483424} a^{11} - \frac{1}{8} a^{10} + \frac{14643215}{166870856} a^{9} - \frac{3}{32} a^{8} + \frac{59576001}{333741712} a^{7} + \frac{1}{8} a^{6} - \frac{24171915}{166870856} a^{5} - \frac{1}{16} a^{4} + \frac{136941561}{667483424} a^{3} - \frac{1}{8} a^{2} + \frac{11510619}{83435428} a - \frac{9}{32}$

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:  $12$
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:  \( 3187004.5686 \) (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

\(\Q(\sqrt{5}) \), 4.2.400.1, 10.6.1749952800000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
5Data not computed
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$