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Properties

Label 20.20.1205670158...8125.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{35}\cdot 23^{10}$
Root discriminant $80.18$
Ramified primes $5, 23$
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![-64286399, 437027220, -520959600, -953469870, 1223413200, 442743929, -720099540, -88063200, 199252740, 8969400, -31120079, -489240, 2948400, 13590, -172800, -151, 6120, 0, -120, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 120*x^18 + 6120*x^16 - 151*x^15 - 172800*x^14 + 13590*x^13 + 2948400*x^12 - 489240*x^11 - 31120079*x^10 + 8969400*x^9 + 199252740*x^8 - 88063200*x^7 - 720099540*x^6 + 442743929*x^5 + 1223413200*x^4 - 953469870*x^3 - 520959600*x^2 + 437027220*x - 64286399)
 
gp: K = bnfinit(x^20 - 120*x^18 + 6120*x^16 - 151*x^15 - 172800*x^14 + 13590*x^13 + 2948400*x^12 - 489240*x^11 - 31120079*x^10 + 8969400*x^9 + 199252740*x^8 - 88063200*x^7 - 720099540*x^6 + 442743929*x^5 + 1223413200*x^4 - 953469870*x^3 - 520959600*x^2 + 437027220*x - 64286399, 1)
 

Normalized defining polynomial

\( x^{20} - 120 x^{18} + 6120 x^{16} - 151 x^{15} - 172800 x^{14} + 13590 x^{13} + 2948400 x^{12} - 489240 x^{11} - 31120079 x^{10} + 8969400 x^{9} + 199252740 x^{8} - 88063200 x^{7} - 720099540 x^{6} + 442743929 x^{5} + 1223413200 x^{4} - 953469870 x^{3} - 520959600 x^{2} + 437027220 x - 64286399 \)

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:  \(120567015877601807005703449249267578125=5^{35}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.18$
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:  $5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(575=5^{2}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{575}(1,·)$, $\chi_{575}(68,·)$, $\chi_{575}(137,·)$, $\chi_{575}(139,·)$, $\chi_{575}(461,·)$, $\chi_{575}(528,·)$, $\chi_{575}(22,·)$, $\chi_{575}(24,·)$, $\chi_{575}(346,·)$, $\chi_{575}(413,·)$, $\chi_{575}(482,·)$, $\chi_{575}(484,·)$, $\chi_{575}(231,·)$, $\chi_{575}(298,·)$, $\chi_{575}(367,·)$, $\chi_{575}(369,·)$, $\chi_{575}(116,·)$, $\chi_{575}(183,·)$, $\chi_{575}(252,·)$, $\chi_{575}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} - \frac{3}{19} a^{8} + \frac{6}{19} a^{6} - \frac{9}{19} a^{5} - \frac{8}{19} a^{4} + \frac{4}{19} a^{3} + \frac{5}{19} a^{2} - \frac{5}{19} a - \frac{5}{19}$, $\frac{1}{19} a^{11} - \frac{3}{19} a^{9} + \frac{6}{19} a^{7} - \frac{9}{19} a^{6} - \frac{8}{19} a^{5} + \frac{4}{19} a^{4} + \frac{5}{19} a^{3} - \frac{5}{19} a^{2} - \frac{5}{19} a$, $\frac{1}{19} a^{12} - \frac{3}{19} a^{8} - \frac{9}{19} a^{7} - \frac{9}{19} a^{6} - \frac{4}{19} a^{5} + \frac{7}{19} a^{3} - \frac{9}{19} a^{2} + \frac{4}{19} a + \frac{4}{19}$, $\frac{1}{920550931} a^{13} - \frac{9403020}{920550931} a^{12} - \frac{78}{920550931} a^{11} - \frac{1283246}{920550931} a^{10} + \frac{2340}{920550931} a^{9} + \frac{83097691}{920550931} a^{8} + \frac{15298242}{48450049} a^{7} + \frac{295580079}{920550931} a^{6} + \frac{339386215}{920550931} a^{5} + \frac{390432882}{920550931} a^{4} - \frac{388308008}{920550931} a^{3} + \frac{340244942}{920550931} a^{2} - \frac{386993864}{920550931} a - \frac{420265291}{920550931}$, $\frac{1}{920550931} a^{14} - \frac{84}{920550931} a^{12} - \frac{7968071}{920550931} a^{11} + \frac{2772}{920550931} a^{10} - \frac{18221484}{48450049} a^{9} - \frac{45360}{920550931} a^{8} + \frac{411638717}{920550931} a^{7} + \frac{387981416}{920550931} a^{6} + \frac{110972955}{920550931} a^{5} + \frac{337626247}{920550931} a^{4} - \frac{32626502}{920550931} a^{3} - \frac{191514052}{920550931} a^{2} + \frac{446132214}{920550931} a - \frac{559872}{920550931}$, $\frac{1}{920550931} a^{15} - \frac{22620967}{920550931} a^{12} - \frac{3780}{920550931} a^{11} - \frac{17950419}{920550931} a^{10} + \frac{151200}{920550931} a^{9} + \frac{75887362}{920550931} a^{8} + \frac{336700903}{920550931} a^{7} + \frac{327074699}{920550931} a^{6} - \frac{272411142}{920550931} a^{5} - \frac{182297734}{920550931} a^{4} + \frac{136646596}{920550931} a^{3} + \frac{295828285}{920550931} a^{2} - \frac{288761863}{920550931} a - \frac{321349066}{920550931}$, $\frac{1}{920550931} a^{16} - \frac{4320}{920550931} a^{12} + \frac{10265968}{920550931} a^{11} + \frac{190080}{920550931} a^{10} - \frac{237653709}{920550931} a^{9} - \frac{3499200}{920550931} a^{8} - \frac{128908906}{920550931} a^{7} - \frac{113997315}{920550931} a^{6} + \frac{5684463}{920550931} a^{5} - \frac{372887045}{920550931} a^{4} + \frac{313282282}{920550931} a^{3} - \frac{186046472}{920550931} a^{2} + \frac{328148180}{920550931} a - \frac{50388480}{920550931}$, $\frac{1}{920550931} a^{17} - \frac{9639370}{920550931} a^{12} - \frac{146880}{920550931} a^{11} - \frac{15720598}{920550931} a^{10} + \frac{6609600}{920550931} a^{9} - \frac{258646974}{920550931} a^{8} - \frac{17313790}{920550931} a^{7} - \frac{231665897}{920550931} a^{6} + \frac{452279799}{920550931} a^{5} + \frac{437126832}{920550931} a^{4} + \frac{293905985}{920550931} a^{3} + \frac{405611156}{920550931} a^{2} - \frac{46490166}{920550931} a - \frac{122721090}{920550931}$, $\frac{1}{920550931} a^{18} - \frac{176256}{920550931} a^{12} + \frac{7609326}{920550931} a^{11} + \frac{8724672}{920550931} a^{10} - \frac{280244008}{920550931} a^{9} - \frac{171320832}{920550931} a^{8} + \frac{447551765}{920550931} a^{7} - \frac{242107430}{920550931} a^{6} - \frac{441392882}{920550931} a^{5} - \frac{408976763}{920550931} a^{4} + \frac{182617407}{920550931} a^{3} - \frac{253398756}{920550931} a^{2} - \frac{42326560}{920550931} a + \frac{20519481}{920550931}$, $\frac{1}{920550931} a^{19} - \frac{257651}{920550931} a^{12} - \frac{264384}{48450049} a^{11} - \frac{4521958}{920550931} a^{10} + \frac{12690432}{48450049} a^{9} - \frac{68586578}{920550931} a^{8} - \frac{221873579}{920550931} a^{7} + \frac{291472965}{920550931} a^{6} - \frac{308363475}{920550931} a^{5} - \frac{384730637}{920550931} a^{4} - \frac{216088277}{920550931} a^{3} - \frac{331480628}{920550931} a^{2} - \frac{392139886}{920550931} a + \frac{128684722}{920550931}$

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:  \( 3975952893490 \) (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

\(\Q(\sqrt{5}) \), 4.4.66125.1, 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
23Data not computed