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Properties

Label 20.0.25948115648...6689.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 41^{19}$
Root discriminant $58.98$
Ramified primes $3, 41$
Class number $544$ (GRH)
Class group $[2, 2, 2, 2, 34]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51907, 37115, 50170, -82739, 74910, 10386, 2671, 16816, -8231, 5692, -1773, 316, -476, -506, 304, -177, 153, -23, 22, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 22*x^18 - 23*x^17 + 153*x^16 - 177*x^15 + 304*x^14 - 506*x^13 - 476*x^12 + 316*x^11 - 1773*x^10 + 5692*x^9 - 8231*x^8 + 16816*x^7 + 2671*x^6 + 10386*x^5 + 74910*x^4 - 82739*x^3 + 50170*x^2 + 37115*x + 51907)
 
gp: K = bnfinit(x^20 - x^19 + 22*x^18 - 23*x^17 + 153*x^16 - 177*x^15 + 304*x^14 - 506*x^13 - 476*x^12 + 316*x^11 - 1773*x^10 + 5692*x^9 - 8231*x^8 + 16816*x^7 + 2671*x^6 + 10386*x^5 + 74910*x^4 - 82739*x^3 + 50170*x^2 + 37115*x + 51907, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 22 x^{18} - 23 x^{17} + 153 x^{16} - 177 x^{15} + 304 x^{14} - 506 x^{13} - 476 x^{12} + 316 x^{11} - 1773 x^{10} + 5692 x^{9} - 8231 x^{8} + 16816 x^{7} + 2671 x^{6} + 10386 x^{5} + 74910 x^{4} - 82739 x^{3} + 50170 x^{2} + 37115 x + 51907 \)

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:  \(259481156482773157712790472689766689=3^{10}\cdot 41^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.98$
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:  $3, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(123=3\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{123}(64,·)$, $\chi_{123}(1,·)$, $\chi_{123}(2,·)$, $\chi_{123}(4,·)$, $\chi_{123}(5,·)$, $\chi_{123}(8,·)$, $\chi_{123}(10,·)$, $\chi_{123}(77,·)$, $\chi_{123}(16,·)$, $\chi_{123}(20,·)$, $\chi_{123}(25,·)$, $\chi_{123}(31,·)$, $\chi_{123}(32,·)$, $\chi_{123}(80,·)$, $\chi_{123}(100,·)$, $\chi_{123}(37,·)$, $\chi_{123}(40,·)$, $\chi_{123}(50,·)$, $\chi_{123}(74,·)$, $\chi_{123}(62,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{6} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{978} a^{17} + \frac{55}{978} a^{16} - \frac{10}{489} a^{15} + \frac{35}{489} a^{14} - \frac{46}{163} a^{13} + \frac{229}{489} a^{12} - \frac{7}{978} a^{11} + \frac{3}{163} a^{10} - \frac{233}{489} a^{9} + \frac{479}{978} a^{8} + \frac{481}{978} a^{7} - \frac{115}{326} a^{6} - \frac{128}{489} a^{5} - \frac{145}{978} a^{4} + \frac{9}{326} a^{3} - \frac{221}{978} a^{2} + \frac{187}{489} a - \frac{43}{489}$, $\frac{1}{978} a^{18} + \frac{26}{489} a^{16} + \frac{29}{978} a^{15} + \frac{56}{489} a^{14} - \frac{5}{489} a^{13} - \frac{95}{978} a^{12} + \frac{77}{978} a^{11} + \frac{11}{978} a^{10} + \frac{29}{978} a^{9} - \frac{218}{489} a^{8} + \frac{421}{978} a^{7} - \frac{176}{489} a^{6} + \frac{203}{489} a^{5} - \frac{74}{489} a^{4} - \frac{239}{978} a^{3} - \frac{58}{163} a^{2} - \frac{281}{978} a - \frac{81}{163}$, $\frac{1}{33157010133754899772879190462185114485606138} a^{19} - \frac{2204762350005826657165826504807380963379}{16578505066877449886439595231092557242803069} a^{18} - \frac{12268340748678427970228210168039671864253}{33157010133754899772879190462185114485606138} a^{17} - \frac{1109447350096786700928189943561969452955511}{33157010133754899772879190462185114485606138} a^{16} - \frac{1190479349604443862950909003265287190955166}{16578505066877449886439595231092557242803069} a^{15} - \frac{1508613017468217543125955445685377674532029}{11052336711251633257626396820728371495202046} a^{14} - \frac{5019349869274524564406160223029477698434707}{33157010133754899772879190462185114485606138} a^{13} - \frac{5443374547505168201307649454604939255517505}{33157010133754899772879190462185114485606138} a^{12} - \frac{7045344535244519207826900395511349698372952}{16578505066877449886439595231092557242803069} a^{11} - \frac{2773542351479507722254736352151459490707493}{16578505066877449886439595231092557242803069} a^{10} - \frac{2772203149509132338492984126843655345721725}{11052336711251633257626396820728371495202046} a^{9} - \frac{3062890676610232124617908641406861395865409}{16578505066877449886439595231092557242803069} a^{8} + \frac{1209336011943787689410844987432528527277407}{5526168355625816628813198410364185747601023} a^{7} + \frac{2225819295662891906280332623266676667792155}{33157010133754899772879190462185114485606138} a^{6} + \frac{3379832639218474408732020171213014405370224}{16578505066877449886439595231092557242803069} a^{5} + \frac{2731283160968908573404114496781877071794941}{11052336711251633257626396820728371495202046} a^{4} + \frac{1528278125568193374485647723861907431305193}{5526168355625816628813198410364185747601023} a^{3} + \frac{796492615409613976103436869932749217854773}{16578505066877449886439595231092557242803069} a^{2} - \frac{5889367945573970781785628256800148178119681}{16578505066877449886439595231092557242803069} a - \frac{2555565672612486834310934324293295255829403}{33157010133754899772879190462185114485606138}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{34}$, which has order $544$ (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:  \( -1 \) (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:  \( 5104264.63655 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.0.620289.1, 5.5.2825761.1, 10.10.327381934393961.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ $20$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41Data not computed