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Properties

Label 20.6.59313783170...0256.4
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{20}\cdot 3^{19}\cdot 13^{5}\cdot 107^{4}$
Root discriminant $27.46$
Ramified primes $2, 3, 13, 107$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![433, 2, -1826, 970, 2825, -2888, -1357, 2974, -766, -980, 845, -218, -58, 106, -82, 46, -16, -8, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 - 8*x^17 - 16*x^16 + 46*x^15 - 82*x^14 + 106*x^13 - 58*x^12 - 218*x^11 + 845*x^10 - 980*x^9 - 766*x^8 + 2974*x^7 - 1357*x^6 - 2888*x^5 + 2825*x^4 + 970*x^3 - 1826*x^2 + 2*x + 433)
 
gp: K = bnfinit(x^20 - 4*x^19 + 10*x^18 - 8*x^17 - 16*x^16 + 46*x^15 - 82*x^14 + 106*x^13 - 58*x^12 - 218*x^11 + 845*x^10 - 980*x^9 - 766*x^8 + 2974*x^7 - 1357*x^6 - 2888*x^5 + 2825*x^4 + 970*x^3 - 1826*x^2 + 2*x + 433, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 10 x^{18} - 8 x^{17} - 16 x^{16} + 46 x^{15} - 82 x^{14} + 106 x^{13} - 58 x^{12} - 218 x^{11} + 845 x^{10} - 980 x^{9} - 766 x^{8} + 2974 x^{7} - 1357 x^{6} - 2888 x^{5} + 2825 x^{4} + 970 x^{3} - 1826 x^{2} + 2 x + 433 \)

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:  \(-59313783170950766384309600256=-\,2^{20}\cdot 3^{19}\cdot 13^{5}\cdot 107^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.46$
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:  $2, 3, 13, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{9} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{117} a^{17} - \frac{5}{117} a^{16} + \frac{19}{117} a^{15} + \frac{17}{117} a^{14} - \frac{4}{117} a^{13} + \frac{8}{117} a^{12} + \frac{7}{117} a^{11} - \frac{8}{117} a^{10} - \frac{56}{117} a^{9} + \frac{20}{117} a^{8} + \frac{50}{117} a^{7} + \frac{29}{117} a^{6} - \frac{56}{117} a^{5} - \frac{11}{117} a^{4} + \frac{4}{9} a^{3} - \frac{16}{117} a^{2} + \frac{5}{117} a + \frac{56}{117}$, $\frac{1}{1521} a^{18} + \frac{4}{1521} a^{17} - \frac{11}{117} a^{16} - \frac{202}{1521} a^{15} + \frac{149}{1521} a^{14} + \frac{245}{1521} a^{13} + \frac{1}{1521} a^{12} + \frac{250}{1521} a^{11} + \frac{67}{1521} a^{10} - \frac{640}{1521} a^{9} + \frac{737}{1521} a^{8} - \frac{730}{1521} a^{7} + \frac{361}{1521} a^{6} - \frac{710}{1521} a^{5} - \frac{476}{1521} a^{4} - \frac{406}{1521} a^{3} - \frac{412}{1521} a^{2} + \frac{452}{1521} a - \frac{248}{507}$, $\frac{1}{54965488715802873} a^{19} + \frac{681240004103}{4228114516600221} a^{18} - \frac{135568708963834}{54965488715802873} a^{17} - \frac{1430406439942361}{18321829571934291} a^{16} + \frac{2411839390743427}{54965488715802873} a^{15} - \frac{18287485483877}{4228114516600221} a^{14} - \frac{627234827474330}{18321829571934291} a^{13} + \frac{8413928510631539}{54965488715802873} a^{12} + \frac{8034858661896797}{54965488715802873} a^{11} + \frac{9201516715771}{964306819575489} a^{10} + \frac{6209314495923664}{54965488715802873} a^{9} + \frac{4385908672128244}{54965488715802873} a^{8} - \frac{3017118352666906}{18321829571934291} a^{7} - \frac{10825148621029375}{54965488715802873} a^{6} - \frac{19271105273905201}{54965488715802873} a^{5} - \frac{1427954295276716}{18321829571934291} a^{4} + \frac{13578827968407289}{54965488715802873} a^{3} + \frac{8708077647029092}{54965488715802873} a^{2} + \frac{13774421111873912}{54965488715802873} a + \frac{289414711550930}{18321829571934291}$

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:  \( 7112232.34608 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.6.38998285028352.1

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

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R $20$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ $20$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.12.13.4$x^{12} - 3 x^{10} + 3 x^{6} - 3 x^{5} + 3 x^{4} - 3 x^{2} - 3$$12$$1$$13$12T36$[5/4, 5/4]_{4}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.5.1$x^{6} - 52$$6$$1$$5$$C_6$$[\ ]_{6}$
107Data not computed