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Properties

Label 20.20.9273875903...5681.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $39.66$
Ramified primes $3, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![331, 4783, -23758, -15587, 116035, 28344, -212737, -39237, 191218, 30691, -96559, -13187, 29302, 3223, -5463, -448, 613, 33, -38, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 38*x^18 + 33*x^17 + 613*x^16 - 448*x^15 - 5463*x^14 + 3223*x^13 + 29302*x^12 - 13187*x^11 - 96559*x^10 + 30691*x^9 + 191218*x^8 - 39237*x^7 - 212737*x^6 + 28344*x^5 + 116035*x^4 - 15587*x^3 - 23758*x^2 + 4783*x + 331)
 
gp: K = bnfinit(x^20 - x^19 - 38*x^18 + 33*x^17 + 613*x^16 - 448*x^15 - 5463*x^14 + 3223*x^13 + 29302*x^12 - 13187*x^11 - 96559*x^10 + 30691*x^9 + 191218*x^8 - 39237*x^7 - 212737*x^6 + 28344*x^5 + 116035*x^4 - 15587*x^3 - 23758*x^2 + 4783*x + 331, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 38 x^{18} + 33 x^{17} + 613 x^{16} - 448 x^{15} - 5463 x^{14} + 3223 x^{13} + 29302 x^{12} - 13187 x^{11} - 96559 x^{10} + 30691 x^{9} + 191218 x^{8} - 39237 x^{7} - 212737 x^{6} + 28344 x^{5} + 116035 x^{4} - 15587 x^{3} - 23758 x^{2} + 4783 x + 331 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(92738759037689478010716606945681=3^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.66$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(231=3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{231}(64,·)$, $\chi_{231}(1,·)$, $\chi_{231}(197,·)$, $\chi_{231}(134,·)$, $\chi_{231}(8,·)$, $\chi_{231}(139,·)$, $\chi_{231}(76,·)$, $\chi_{231}(13,·)$, $\chi_{231}(146,·)$, $\chi_{231}(20,·)$, $\chi_{231}(29,·)$, $\chi_{231}(160,·)$, $\chi_{231}(104,·)$, $\chi_{231}(169,·)$, $\chi_{231}(50,·)$, $\chi_{231}(118,·)$, $\chi_{231}(148,·)$, $\chi_{231}(188,·)$, $\chi_{231}(125,·)$, $\chi_{231}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{23} a^{11} + \frac{1}{23} a^{9} - \frac{8}{23} a^{7} + \frac{5}{23} a^{5} + \frac{6}{23} a^{3} - \frac{7}{23} a - \frac{1}{23}$, $\frac{1}{23} a^{12} + \frac{1}{23} a^{10} - \frac{8}{23} a^{8} + \frac{5}{23} a^{6} + \frac{6}{23} a^{4} - \frac{7}{23} a^{2} - \frac{1}{23} a$, $\frac{1}{23} a^{13} - \frac{9}{23} a^{9} - \frac{10}{23} a^{7} + \frac{1}{23} a^{5} + \frac{10}{23} a^{3} - \frac{1}{23} a^{2} + \frac{7}{23} a + \frac{1}{23}$, $\frac{1}{23} a^{14} - \frac{9}{23} a^{10} - \frac{10}{23} a^{8} + \frac{1}{23} a^{6} + \frac{10}{23} a^{4} - \frac{1}{23} a^{3} + \frac{7}{23} a^{2} + \frac{1}{23} a$, $\frac{1}{23} a^{15} - \frac{1}{23} a^{9} - \frac{2}{23} a^{7} + \frac{9}{23} a^{5} - \frac{1}{23} a^{4} - \frac{8}{23} a^{3} + \frac{1}{23} a^{2} + \frac{6}{23} a - \frac{9}{23}$, $\frac{1}{56143} a^{16} - \frac{948}{56143} a^{15} + \frac{231}{56143} a^{14} + \frac{1143}{56143} a^{13} + \frac{535}{56143} a^{12} + \frac{70}{56143} a^{11} - \frac{3983}{56143} a^{10} + \frac{287}{2441} a^{9} - \frac{7420}{56143} a^{8} + \frac{8467}{56143} a^{7} - \frac{16428}{56143} a^{6} - \frac{8328}{56143} a^{5} - \frac{13619}{56143} a^{4} + \frac{1816}{56143} a^{3} + \frac{21294}{56143} a^{2} + \frac{16276}{56143} a + \frac{8938}{56143}$, $\frac{1}{56143} a^{17} - \frac{185}{56143} a^{15} + \frac{441}{56143} a^{14} + \frac{295}{56143} a^{13} - \frac{478}{56143} a^{12} - \frac{1089}{56143} a^{11} + \frac{2062}{56143} a^{10} - \frac{3514}{56143} a^{9} + \frac{19033}{56143} a^{8} - \frac{13279}{56143} a^{7} + \frac{18359}{56143} a^{6} + \frac{10041}{56143} a^{5} - \frac{18075}{56143} a^{4} + \frac{7311}{56143} a^{3} + \frac{3713}{56143} a^{2} - \frac{5621}{56143} a - \frac{14133}{56143}$, $\frac{1}{56143} a^{18} + \frac{813}{56143} a^{15} - \frac{908}{56143} a^{14} + \frac{1051}{56143} a^{13} + \frac{246}{56143} a^{12} + \frac{366}{56143} a^{11} + \frac{6577}{56143} a^{10} + \frac{19718}{56143} a^{9} + \frac{2950}{56143} a^{8} + \frac{1085}{2441} a^{7} - \frac{24268}{56143} a^{6} + \frac{20572}{56143} a^{5} - \frac{7738}{56143} a^{4} - \frac{9390}{56143} a^{3} - \frac{20651}{56143} a^{2} - \frac{12826}{56143} a - \frac{11232}{56143}$, $\frac{1}{56143} a^{19} + \frac{901}{56143} a^{15} + \frac{1205}{56143} a^{14} + \frac{1008}{56143} a^{13} - \frac{91}{56143} a^{12} + \frac{928}{56143} a^{11} - \frac{20366}{56143} a^{10} - \frac{786}{56143} a^{9} - \frac{27998}{56143} a^{8} + \frac{22060}{56143} a^{7} + \frac{16943}{56143} a^{6} + \frac{18443}{56143} a^{5} - \frac{19283}{56143} a^{4} - \frac{17813}{56143} a^{3} + \frac{23339}{56143} a^{2} + \frac{6128}{56143} a + \frac{27114}{56143}$

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:  $19$
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:  \( 2904089575.06 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{77}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{21}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 10.10.39630026842637.1, 10.10.875463320250981.1, \(\Q(\zeta_{33})^+\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
7Data not computed
11Data not computed