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Properties

Label 20.4.16452808797...2016.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{16}\cdot 23^{2}\cdot 661$
Root discriminant $18.23$
Ramified primes $2, 11, 23, 661$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T846

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, 160, -224, 32, 0, 24, 172, -136, -20, 228, -37, -147, 61, 61, -36, -31, 11, 8, -5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 5*x^18 + 8*x^17 + 11*x^16 - 31*x^15 - 36*x^14 + 61*x^13 + 61*x^12 - 147*x^11 - 37*x^10 + 228*x^9 - 20*x^8 - 136*x^7 + 172*x^6 + 24*x^5 + 32*x^3 - 224*x^2 + 160*x - 32)
 
gp: K = bnfinit(x^20 - x^19 - 5*x^18 + 8*x^17 + 11*x^16 - 31*x^15 - 36*x^14 + 61*x^13 + 61*x^12 - 147*x^11 - 37*x^10 + 228*x^9 - 20*x^8 - 136*x^7 + 172*x^6 + 24*x^5 + 32*x^3 - 224*x^2 + 160*x - 32, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 5 x^{18} + 8 x^{17} + 11 x^{16} - 31 x^{15} - 36 x^{14} + 61 x^{13} + 61 x^{12} - 147 x^{11} - 37 x^{10} + 228 x^{9} - 20 x^{8} - 136 x^{7} + 172 x^{6} + 24 x^{5} + 32 x^{3} - 224 x^{2} + 160 x - 32 \)

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:  \(16452808797865383899862016=2^{10}\cdot 11^{16}\cdot 23^{2}\cdot 661\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.23$
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:  $2, 11, 23, 661$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{1}{2} a^{10} - \frac{3}{8} a^{9} - \frac{3}{8} a^{8} - \frac{3}{8} a^{7} + \frac{3}{8} a^{6}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{4} a^{12} + \frac{3}{8} a^{11} - \frac{3}{8} a^{10} - \frac{1}{2} a^{8} + \frac{3}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{18} - \frac{1}{16} a^{17} - \frac{1}{16} a^{16} - \frac{1}{16} a^{14} + \frac{1}{16} a^{13} - \frac{1}{4} a^{12} - \frac{3}{16} a^{11} - \frac{3}{16} a^{10} + \frac{5}{16} a^{9} + \frac{3}{16} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2233978240825028101168} a^{19} + \frac{4976515958925674691}{2233978240825028101168} a^{18} + \frac{123607711143557345509}{2233978240825028101168} a^{17} + \frac{47277905185242079963}{1116989120412514050584} a^{16} - \frac{72159402585321393435}{2233978240825028101168} a^{15} + \frac{181324460802411770673}{2233978240825028101168} a^{14} + \frac{91656469658798825051}{1116989120412514050584} a^{13} - \frac{80769812204302745889}{2233978240825028101168} a^{12} + \frac{374291652689869664545}{2233978240825028101168} a^{11} - \frac{1039813717022067203381}{2233978240825028101168} a^{10} + \frac{161547502917043646489}{2233978240825028101168} a^{9} + \frac{480876604629405482175}{1116989120412514050584} a^{8} + \frac{338948070147189862537}{1116989120412514050584} a^{7} - \frac{58825916247481778807}{558494560206257025292} a^{6} - \frac{39233019409910095210}{139623640051564256323} a^{5} + \frac{221413139827783349143}{558494560206257025292} a^{4} - \frac{9681562663888163698}{139623640051564256323} a^{3} + \frac{131644364214409416765}{279247280103128512646} a^{2} - \frac{53712716400840693554}{139623640051564256323} a - \frac{67701427514225867038}{139623640051564256323}$

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:  \( 40135.725167 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T846:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 163840
The 649 conjugacy class representatives for t20n846 are not computed
Character table for t20n846 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.4930254263.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
661Data not computed