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Properties

Label 20.20.2133342346...8125.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{15}\cdot 31^{18}$
Root discriminant $73.53$
Ramified primes $5, 31$
Class number $1$ (GRH)
Class group Trivial (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![-4645, 8405, 63070, -137680, -146594, 478381, -12337, -586336, 205983, 341431, -174550, -104480, 64636, 17240, -12200, -1526, 1198, 66, -57, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 57*x^18 + 66*x^17 + 1198*x^16 - 1526*x^15 - 12200*x^14 + 17240*x^13 + 64636*x^12 - 104480*x^11 - 174550*x^10 + 341431*x^9 + 205983*x^8 - 586336*x^7 - 12337*x^6 + 478381*x^5 - 146594*x^4 - 137680*x^3 + 63070*x^2 + 8405*x - 4645)
 
gp: K = bnfinit(x^20 - x^19 - 57*x^18 + 66*x^17 + 1198*x^16 - 1526*x^15 - 12200*x^14 + 17240*x^13 + 64636*x^12 - 104480*x^11 - 174550*x^10 + 341431*x^9 + 205983*x^8 - 586336*x^7 - 12337*x^6 + 478381*x^5 - 146594*x^4 - 137680*x^3 + 63070*x^2 + 8405*x - 4645, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 57 x^{18} + 66 x^{17} + 1198 x^{16} - 1526 x^{15} - 12200 x^{14} + 17240 x^{13} + 64636 x^{12} - 104480 x^{11} - 174550 x^{10} + 341431 x^{9} + 205983 x^{8} - 586336 x^{7} - 12337 x^{6} + 478381 x^{5} - 146594 x^{4} - 137680 x^{3} + 63070 x^{2} + 8405 x - 4645 \)

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:  \(21333423461884919389012763702392578125=5^{15}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.53$
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:  $5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(155=5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{155}(64,·)$, $\chi_{155}(1,·)$, $\chi_{155}(66,·)$, $\chi_{155}(4,·)$, $\chi_{155}(77,·)$, $\chi_{155}(16,·)$, $\chi_{155}(147,·)$, $\chi_{155}(23,·)$, $\chi_{155}(153,·)$, $\chi_{155}(27,·)$, $\chi_{155}(92,·)$, $\chi_{155}(58,·)$, $\chi_{155}(94,·)$, $\chi_{155}(101,·)$, $\chi_{155}(39,·)$, $\chi_{155}(108,·)$, $\chi_{155}(109,·)$, $\chi_{155}(122,·)$, $\chi_{155}(123,·)$, $\chi_{155}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{15} + \frac{1}{5} a^{12} + \frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{15} + \frac{1}{5} a^{13} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4}$, $\frac{1}{3095} a^{18} - \frac{168}{3095} a^{17} + \frac{57}{3095} a^{16} + \frac{1449}{3095} a^{15} - \frac{1239}{3095} a^{14} + \frac{266}{619} a^{13} + \frac{378}{3095} a^{12} - \frac{255}{619} a^{11} + \frac{1408}{3095} a^{10} - \frac{56}{619} a^{9} - \frac{1448}{3095} a^{8} + \frac{368}{3095} a^{7} - \frac{1}{5} a^{6} - \frac{1209}{3095} a^{5} + \frac{152}{3095} a^{4} + \frac{216}{619} a^{3} + \frac{302}{619} a^{2} - \frac{198}{619} a + \frac{217}{619}$, $\frac{1}{1142770404775246457728326362056225} a^{19} - \frac{147644369169537332652980171604}{1142770404775246457728326362056225} a^{18} + \frac{4191839919646701769831380530146}{45710816191009858309133054482249} a^{17} - \frac{73846108351329951030909952493034}{1142770404775246457728326362056225} a^{16} - \frac{103949715341984625420229535174157}{228554080955049291545665272411245} a^{15} - \frac{186982074240470167769859427392581}{1142770404775246457728326362056225} a^{14} - \frac{128118581724046315988410467596712}{1142770404775246457728326362056225} a^{13} + \frac{155653287309513488501469391339311}{1142770404775246457728326362056225} a^{12} - \frac{398969024708578000435834407617407}{1142770404775246457728326362056225} a^{11} + \frac{524162513247643574281854356484501}{1142770404775246457728326362056225} a^{10} - \frac{302723380704464937364247858082013}{1142770404775246457728326362056225} a^{9} + \frac{12521537923109280680112075591351}{228554080955049291545665272411245} a^{8} + \frac{217132950119565331880919781356388}{1142770404775246457728326362056225} a^{7} + \frac{97051421677546301796666515193697}{228554080955049291545665272411245} a^{6} - \frac{503204555778300340552523269516842}{1142770404775246457728326362056225} a^{5} - \frac{443017724419012442083711477126598}{1142770404775246457728326362056225} a^{4} - \frac{13446756289583722653674067515676}{228554080955049291545665272411245} a^{3} + \frac{50676870164432960674516785171232}{228554080955049291545665272411245} a^{2} - \frac{43499705781688316794821034280967}{228554080955049291545665272411245} a - \frac{60561366972531604526174246059473}{228554080955049291545665272411245}$

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:  \( 3696529687160 \) (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

\(\Q(\sqrt{5}) \), 4.4.120125.1, 5.5.923521.1, 10.10.2665284492003125.1

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 $20$ $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$31$31.10.9.8$x^{10} + 521017$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.9.8$x^{10} + 521017$$10$$1$$9$$C_{10}$$[\ ]_{10}$