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Properties

Label 20.0.21822182114...9209.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 883^{8}$
Root discriminant $26.12$
Ramified primes $3, 883$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -1536, 2304, 768, -896, 1248, 896, -456, 496, 402, -181, 201, 124, -57, 56, 39, -14, 6, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 9*x^18 + 6*x^17 - 14*x^16 + 39*x^15 + 56*x^14 - 57*x^13 + 124*x^12 + 201*x^11 - 181*x^10 + 402*x^9 + 496*x^8 - 456*x^7 + 896*x^6 + 1248*x^5 - 896*x^4 + 768*x^3 + 2304*x^2 - 1536*x + 1024)
 
gp: K = bnfinit(x^20 - 3*x^19 + 9*x^18 + 6*x^17 - 14*x^16 + 39*x^15 + 56*x^14 - 57*x^13 + 124*x^12 + 201*x^11 - 181*x^10 + 402*x^9 + 496*x^8 - 456*x^7 + 896*x^6 + 1248*x^5 - 896*x^4 + 768*x^3 + 2304*x^2 - 1536*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 9 x^{18} + 6 x^{17} - 14 x^{16} + 39 x^{15} + 56 x^{14} - 57 x^{13} + 124 x^{12} + 201 x^{11} - 181 x^{10} + 402 x^{9} + 496 x^{8} - 456 x^{7} + 896 x^{6} + 1248 x^{5} - 896 x^{4} + 768 x^{3} + 2304 x^{2} - 1536 x + 1024 \)

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:  \(21822182114965484993955979209=3^{10}\cdot 883^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.12$
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:  $3, 883$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a$, $\frac{1}{36} a^{12} - \frac{1}{12} a^{11} + \frac{5}{36} a^{10} - \frac{1}{6} a^{9} + \frac{1}{18} a^{8} + \frac{1}{12} a^{7} - \frac{4}{9} a^{6} - \frac{1}{4} a^{5} + \frac{2}{9} a^{4} + \frac{1}{4} a^{3} - \frac{1}{36} a^{2} + \frac{1}{6} a + \frac{1}{9}$, $\frac{1}{72} a^{13} - \frac{1}{72} a^{12} - \frac{1}{72} a^{11} + \frac{1}{18} a^{10} + \frac{1}{36} a^{9} - \frac{5}{72} a^{8} - \frac{5}{36} a^{7} - \frac{5}{72} a^{6} + \frac{1}{36} a^{5} + \frac{13}{72} a^{4} - \frac{19}{72} a^{3} + \frac{1}{18} a^{2} - \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{432} a^{14} - \frac{1}{144} a^{13} - \frac{1}{144} a^{12} - \frac{5}{72} a^{11} + \frac{23}{216} a^{10} + \frac{13}{144} a^{9} + \frac{1}{27} a^{8} - \frac{5}{48} a^{7} + \frac{1}{108} a^{6} + \frac{71}{144} a^{5} + \frac{163}{432} a^{4} - \frac{7}{72} a^{3} + \frac{11}{36} a^{2} + \frac{5}{18} a - \frac{8}{27}$, $\frac{1}{864} a^{15} - \frac{1}{864} a^{14} + \frac{1}{288} a^{13} - \frac{13}{432} a^{11} - \frac{13}{864} a^{10} + \frac{59}{432} a^{9} - \frac{121}{864} a^{8} - \frac{175}{432} a^{7} + \frac{257}{864} a^{6} - \frac{395}{864} a^{5} - \frac{23}{108} a^{4} - \frac{1}{24} a^{3} - \frac{1}{18} a^{2} + \frac{5}{27} a - \frac{8}{27}$, $\frac{1}{1728} a^{16} - \frac{1}{1728} a^{15} - \frac{1}{1728} a^{14} - \frac{1}{144} a^{13} + \frac{5}{864} a^{12} + \frac{131}{1728} a^{11} - \frac{3}{32} a^{10} - \frac{37}{1728} a^{9} + \frac{47}{288} a^{8} - \frac{187}{1728} a^{7} + \frac{191}{576} a^{6} - \frac{199}{432} a^{5} - \frac{53}{216} a^{4} - \frac{1}{6} a^{3} + \frac{25}{108} a^{2} + \frac{19}{54} a + \frac{5}{27}$, $\frac{1}{3456} a^{17} - \frac{1}{3456} a^{16} - \frac{1}{3456} a^{15} - \frac{1}{864} a^{14} - \frac{7}{1728} a^{13} + \frac{11}{3456} a^{12} - \frac{19}{576} a^{11} + \frac{427}{3456} a^{10} - \frac{31}{192} a^{9} - \frac{251}{3456} a^{8} + \frac{551}{1152} a^{7} + \frac{337}{864} a^{6} + \frac{13}{108} a^{5} + \frac{31}{432} a^{4} + \frac{29}{108} a^{3} - \frac{35}{108} a^{2} - \frac{8}{27} a - \frac{11}{27}$, $\frac{1}{20736} a^{18} - \frac{1}{6912} a^{17} + \frac{5}{20736} a^{16} - \frac{1}{3456} a^{15} - \frac{5}{10368} a^{14} - \frac{35}{6912} a^{13} + \frac{1}{108} a^{12} + \frac{233}{6912} a^{11} - \frac{659}{5184} a^{10} - \frac{977}{6912} a^{9} + \frac{2995}{20736} a^{8} + \frac{127}{1152} a^{7} + \frac{265}{1728} a^{6} - \frac{5}{96} a^{5} + \frac{299}{1296} a^{4} + \frac{5}{36} a^{3} - \frac{91}{324} a^{2} + \frac{7}{54} a - \frac{11}{81}$, $\frac{1}{290304} a^{19} + \frac{1}{290304} a^{18} + \frac{41}{290304} a^{17} - \frac{41}{145152} a^{16} - \frac{17}{145152} a^{15} - \frac{97}{290304} a^{14} + \frac{13}{2688} a^{13} + \frac{313}{96768} a^{12} - \frac{797}{18144} a^{11} - \frac{4019}{290304} a^{10} + \frac{12919}{290304} a^{9} + \frac{9701}{145152} a^{8} - \frac{8077}{24192} a^{7} - \frac{1807}{4032} a^{6} + \frac{5645}{18144} a^{5} - \frac{1255}{4536} a^{4} - \frac{2185}{4536} a^{3} + \frac{277}{2268} a^{2} - \frac{95}{1134} a + \frac{38}{567}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( -\frac{1051}{32256} a^{19} + \frac{815}{3584} a^{18} - \frac{12557}{32256} a^{17} + \frac{2831}{8064} a^{16} + \frac{47771}{16128} a^{15} + \frac{2327}{3584} a^{14} + \frac{3595}{5376} a^{13} + \frac{393707}{32256} a^{12} + \frac{88507}{16128} a^{11} + \frac{58829}{32256} a^{10} + \frac{1153421}{32256} a^{9} + \frac{100013}{8064} a^{8} + \frac{24499}{2688} a^{7} + \frac{399079}{4032} a^{6} + \frac{84193}{2016} a^{5} + \frac{38}{21} a^{4} + \frac{92107}{504} a^{3} + \frac{7909}{63} a^{2} - \frac{4877}{63} a + \frac{2029}{21} \) (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:  \( 2605089.85061 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 24 conjugacy class representatives for t20n230
Character table for t20n230 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.7017201.1, 10.0.147723329623203.1, 10.8.147723329623203.1, 10.2.49241109874401.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
883Data not computed