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Properties

Label 20.10.2906949957...0656.3
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{30}\cdot 3^{10}\cdot 71^{9}$
Root discriminant $33.36$
Ramified primes $2, 3, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T647

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -22, 150, -348, -118, 1630, -1440, -1944, 3091, -352, -1404, 1910, -1518, 120, 296, 40, -102, 12, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 18*x^18 + 12*x^17 - 102*x^16 + 40*x^15 + 296*x^14 + 120*x^13 - 1518*x^12 + 1910*x^11 - 1404*x^10 - 352*x^9 + 3091*x^8 - 1944*x^7 - 1440*x^6 + 1630*x^5 - 118*x^4 - 348*x^3 + 150*x^2 - 22*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 18*x^18 + 12*x^17 - 102*x^16 + 40*x^15 + 296*x^14 + 120*x^13 - 1518*x^12 + 1910*x^11 - 1404*x^10 - 352*x^9 + 3091*x^8 - 1944*x^7 - 1440*x^6 + 1630*x^5 - 118*x^4 - 348*x^3 + 150*x^2 - 22*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 18 x^{18} + 12 x^{17} - 102 x^{16} + 40 x^{15} + 296 x^{14} + 120 x^{13} - 1518 x^{12} + 1910 x^{11} - 1404 x^{10} - 352 x^{9} + 3091 x^{8} - 1944 x^{7} - 1440 x^{6} + 1630 x^{5} - 118 x^{4} - 348 x^{3} + 150 x^{2} - 22 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2906949957743139152698231750656=-\,2^{30}\cdot 3^{10}\cdot 71^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.36$
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:  $2, 3, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{43742184135369647484857874613} a^{19} + \frac{9165753285209100682464976622}{43742184135369647484857874613} a^{18} + \frac{2262437461752063954815592008}{43742184135369647484857874613} a^{17} + \frac{1781511331735712811880393990}{43742184135369647484857874613} a^{16} + \frac{718580191892042431299650685}{3364783395028434421912144201} a^{15} - \frac{1269164648295343708150756266}{43742184135369647484857874613} a^{14} - \frac{921363972729021355881347649}{3364783395028434421912144201} a^{13} + \frac{19430679017490462772751098284}{43742184135369647484857874613} a^{12} - \frac{9706008319359881811575417528}{43742184135369647484857874613} a^{11} + \frac{4362326646059123395575400813}{43742184135369647484857874613} a^{10} + \frac{1748679874620370148512107183}{43742184135369647484857874613} a^{9} + \frac{21046196869347908873399837171}{43742184135369647484857874613} a^{8} + \frac{6552380226618852069734594892}{43742184135369647484857874613} a^{7} + \frac{10418413431873814179378054712}{43742184135369647484857874613} a^{6} + \frac{7391598681369882465906301839}{43742184135369647484857874613} a^{5} + \frac{4349341119097939769700827574}{43742184135369647484857874613} a^{4} - \frac{1935823632159236536771964555}{43742184135369647484857874613} a^{3} - \frac{18821638981129003289188745344}{43742184135369647484857874613} a^{2} - \frac{153352355623662888344505804}{3364783395028434421912144201} a + \frac{14809390135731919998696517693}{43742184135369647484857874613}$

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:  $14$
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:  \( 81132481.0588 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T647:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 51200
The 152 conjugacy class representatives for t20n647 are not computed
Character table for t20n647 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.10.6323239406592.1

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$71$71.10.0.1$x^{10} - x + 22$$1$$10$$0$$C_{10}$$[\ ]^{10}$
71.10.9.5$x^{10} - 18176$$10$$1$$9$$C_{10}$$[\ ]_{10}$