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Properties

Label 20.0.47644389152...2121.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{18}\cdot 223^{6}$
Root discriminant $13.61$
Ramified primes $3, 223$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T288

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 16, -33, 57, -66, 54, -15, -36, 78, -99, 78, -36, -15, 54, -66, 57, -33, 16, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 16*x^18 - 33*x^17 + 57*x^16 - 66*x^15 + 54*x^14 - 15*x^13 - 36*x^12 + 78*x^11 - 99*x^10 + 78*x^9 - 36*x^8 - 15*x^7 + 54*x^6 - 66*x^5 + 57*x^4 - 33*x^3 + 16*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 16*x^18 - 33*x^17 + 57*x^16 - 66*x^15 + 54*x^14 - 15*x^13 - 36*x^12 + 78*x^11 - 99*x^10 + 78*x^9 - 36*x^8 - 15*x^7 + 54*x^6 - 66*x^5 + 57*x^4 - 33*x^3 + 16*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 16 x^{18} - 33 x^{17} + 57 x^{16} - 66 x^{15} + 54 x^{14} - 15 x^{13} - 36 x^{12} + 78 x^{11} - 99 x^{10} + 78 x^{9} - 36 x^{8} - 15 x^{7} + 54 x^{6} - 66 x^{5} + 57 x^{4} - 33 x^{3} + 16 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(47644389152686853402121=3^{18}\cdot 223^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.61$
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:  $3, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{425827} a^{18} - \frac{23024}{425827} a^{17} - \frac{188168}{425827} a^{16} + \frac{187598}{425827} a^{15} - \frac{211}{1583} a^{14} - \frac{146238}{425827} a^{13} + \frac{662}{425827} a^{12} - \frac{187748}{425827} a^{11} - \frac{135457}{425827} a^{10} - \frac{194069}{425827} a^{9} - \frac{135457}{425827} a^{8} - \frac{187748}{425827} a^{7} + \frac{662}{425827} a^{6} - \frac{146238}{425827} a^{5} - \frac{211}{1583} a^{4} + \frac{187598}{425827} a^{3} - \frac{188168}{425827} a^{2} - \frac{23024}{425827} a + \frac{1}{425827}$, $\frac{1}{5535751} a^{19} + \frac{6}{5535751} a^{18} + \frac{2278689}{5535751} a^{17} - \frac{531717}{5535751} a^{16} - \frac{115561}{5535751} a^{15} - \frac{17118}{5535751} a^{14} + \frac{431092}{5535751} a^{13} + \frac{579994}{5535751} a^{12} - \frac{1402020}{5535751} a^{11} - \frac{160177}{5535751} a^{10} + \frac{2490627}{5535751} a^{9} - \frac{2282991}{5535751} a^{8} - \frac{840074}{5535751} a^{7} + \frac{1047331}{5535751} a^{6} - \frac{1329637}{5535751} a^{5} + \frac{1594199}{5535751} a^{4} - \frac{2376105}{5535751} a^{3} + \frac{535142}{5535751} a^{2} + \frac{1615204}{5535751} a + \frac{2577992}{5535751}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{68925}{20579} a^{19} + \frac{385235}{20579} a^{18} - \frac{945974}{20579} a^{17} + \frac{1890664}{20579} a^{16} - \frac{3153167}{20579} a^{15} + \frac{3248859}{20579} a^{14} - \frac{2372033}{20579} a^{13} - \frac{2471}{20579} a^{12} + \frac{2549525}{20579} a^{11} - \frac{4377065}{20579} a^{10} + \frac{5032595}{20579} a^{9} - \frac{3272213}{20579} a^{8} + \frac{1066688}{20579} a^{7} + \frac{1556242}{20579} a^{6} - \frac{3139562}{20579} a^{5} + \frac{3272148}{20579} a^{4} - \frac{2553503}{20579} a^{3} + \frac{1167046}{20579} a^{2} - \frac{566885}{20579} a + \frac{161838}{20579} \) (order $6$)
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:  \( 3133.07637196 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T288:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n288
Character table for t20n288 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.3.18063.1, 10.0.978815907.1, 10.0.72758649087.1, 10.6.218275947261.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
3Data not computed
223Data not computed