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Properties

Label 20.0.40122101737...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{35}\cdot 13^{10}$
Root discriminant $60.28$
Ramified primes $5, 13$
Class number $15362$ (GRH)
Class group $[15362]$ (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![18662101, -13549320, 9995400, -10075065, 10764225, -5859721, 7022340, -2223450, 3246330, -452925, 976936, -49410, 184275, -2745, 21600, -61, 1530, 0, 60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 60*x^18 + 1530*x^16 - 61*x^15 + 21600*x^14 - 2745*x^13 + 184275*x^12 - 49410*x^11 + 976936*x^10 - 452925*x^9 + 3246330*x^8 - 2223450*x^7 + 7022340*x^6 - 5859721*x^5 + 10764225*x^4 - 10075065*x^3 + 9995400*x^2 - 13549320*x + 18662101)
 
gp: K = bnfinit(x^20 + 60*x^18 + 1530*x^16 - 61*x^15 + 21600*x^14 - 2745*x^13 + 184275*x^12 - 49410*x^11 + 976936*x^10 - 452925*x^9 + 3246330*x^8 - 2223450*x^7 + 7022340*x^6 - 5859721*x^5 + 10764225*x^4 - 10075065*x^3 + 9995400*x^2 - 13549320*x + 18662101, 1)
 

Normalized defining polynomial

\( x^{20} + 60 x^{18} + 1530 x^{16} - 61 x^{15} + 21600 x^{14} - 2745 x^{13} + 184275 x^{12} - 49410 x^{11} + 976936 x^{10} - 452925 x^{9} + 3246330 x^{8} - 2223450 x^{7} + 7022340 x^{6} - 5859721 x^{5} + 10764225 x^{4} - 10075065 x^{3} + 9995400 x^{2} - 13549320 x + 18662101 \)

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:  \(401221017379430122673511505126953125=5^{35}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.28$
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:  $5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(325=5^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{325}(1,·)$, $\chi_{325}(66,·)$, $\chi_{325}(131,·)$, $\chi_{325}(196,·)$, $\chi_{325}(261,·)$, $\chi_{325}(12,·)$, $\chi_{325}(77,·)$, $\chi_{325}(14,·)$, $\chi_{325}(79,·)$, $\chi_{325}(144,·)$, $\chi_{325}(209,·)$, $\chi_{325}(274,·)$, $\chi_{325}(142,·)$, $\chi_{325}(207,·)$, $\chi_{325}(272,·)$, $\chi_{325}(38,·)$, $\chi_{325}(103,·)$, $\chi_{325}(168,·)$, $\chi_{325}(233,·)$, $\chi_{325}(298,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} - \frac{8}{19} a^{8} - \frac{8}{19} a^{6} - \frac{2}{19} a^{5} + \frac{1}{19} a^{4} + \frac{8}{19} a^{3} - \frac{8}{19} a^{2} + \frac{5}{19} a - \frac{4}{19}$, $\frac{1}{19} a^{11} - \frac{8}{19} a^{9} - \frac{8}{19} a^{7} - \frac{2}{19} a^{6} + \frac{1}{19} a^{5} + \frac{8}{19} a^{4} - \frac{8}{19} a^{3} + \frac{5}{19} a^{2} - \frac{4}{19} a$, $\frac{1}{19} a^{12} + \frac{4}{19} a^{8} - \frac{2}{19} a^{7} - \frac{6}{19} a^{6} - \frac{8}{19} a^{5} - \frac{7}{19} a^{3} + \frac{8}{19} a^{2} + \frac{2}{19} a + \frac{6}{19}$, $\frac{1}{315732481} a^{13} - \frac{7415831}{315732481} a^{12} + \frac{39}{315732481} a^{11} - \frac{1089932}{315732481} a^{10} + \frac{585}{315732481} a^{9} + \frac{18520916}{315732481} a^{8} - \frac{49848285}{315732481} a^{7} + \frac{91238201}{315732481} a^{6} + \frac{14742}{315732481} a^{5} - \frac{8233950}{315732481} a^{4} + \frac{132962105}{315732481} a^{3} + \frac{17644007}{315732481} a^{2} - \frac{132930515}{315732481} a + \frac{6422776}{16617499}$, $\frac{1}{315732481} a^{14} + \frac{42}{315732481} a^{12} + \frac{5629994}{315732481} a^{11} + \frac{693}{315732481} a^{10} - \frac{13620186}{315732481} a^{9} + \frac{5670}{315732481} a^{8} + \frac{2732758}{315732481} a^{7} - \frac{132916178}{315732481} a^{6} - \frac{126901762}{315732481} a^{5} - \frac{99657366}{315732481} a^{4} - \frac{44035218}{315732481} a^{3} + \frac{16653220}{315732481} a^{2} + \frac{26754866}{315732481} a + \frac{4374}{315732481}$, $\frac{1}{315732481} a^{15} + \frac{1362415}{315732481} a^{12} - \frac{945}{315732481} a^{11} - \frac{1078040}{315732481} a^{10} - \frac{18900}{315732481} a^{9} + \frac{122199232}{315732481} a^{8} + \frac{66316906}{315732481} a^{7} + \frac{95763552}{315732481} a^{6} - \frac{33806534}{315732481} a^{5} - \frac{47176797}{315732481} a^{4} - \frac{150450516}{315732481} a^{3} - \frac{132680963}{315732481} a^{2} + \frac{49458837}{315732481} a + \frac{59284440}{315732481}$, $\frac{1}{315732481} a^{16} - \frac{1080}{315732481} a^{12} - \frac{4359728}{315732481} a^{11} - \frac{23760}{315732481} a^{10} - \frac{126436076}{315732481} a^{9} - \frac{218700}{315732481} a^{8} + \frac{7221232}{315732481} a^{7} - \frac{150537267}{315732481} a^{6} - \frac{91194933}{315732481} a^{5} - \frac{1856642}{16617499} a^{4} + \frac{67171817}{315732481} a^{3} - \frac{101279634}{315732481} a^{2} + \frac{92709206}{315732481} a - \frac{196830}{315732481}$, $\frac{1}{315732481} a^{17} - \frac{3822690}{315732481} a^{12} + \frac{18360}{315732481} a^{11} - \frac{7397714}{315732481} a^{10} + \frac{413100}{315732481} a^{9} + \frac{2341716}{315732481} a^{8} + \frac{86656679}{315732481} a^{7} - \frac{79089424}{315732481} a^{6} - \frac{69207335}{315732481} a^{5} + \frac{48250283}{315732481} a^{4} - \frac{77397594}{315732481} a^{3} - \frac{78209577}{315732481} a^{2} - \frac{139519161}{315732481} a + \frac{68448947}{315732481}$, $\frac{1}{315732481} a^{18} + \frac{22032}{315732481} a^{12} - \frac{7870295}{315732481} a^{11} + \frac{545292}{315732481} a^{10} - \frac{37981997}{315732481} a^{9} + \frac{5353776}{315732481} a^{8} + \frac{152369311}{315732481} a^{7} + \frac{7437199}{16617499} a^{6} + \frac{19172156}{315732481} a^{5} - \frac{96019731}{315732481} a^{4} + \frac{52656972}{315732481} a^{3} - \frac{90779006}{315732481} a^{2} - \frac{112858232}{315732481} a + \frac{5353776}{315732481}$, $\frac{1}{315732481} a^{19} - \frac{5531871}{315732481} a^{12} - \frac{16524}{16617499} a^{11} - \frac{192170}{16617499} a^{10} - \frac{396576}{16617499} a^{9} - \frac{73791543}{315732481} a^{8} - \frac{51196997}{315732481} a^{7} - \frac{54958238}{315732481} a^{6} + \frac{110944493}{315732481} a^{5} - \frac{66515704}{315732481} a^{4} + \frac{86727338}{315732481} a^{3} + \frac{70326140}{315732481} a^{2} + \frac{95671494}{315732481} a + \frac{135774889}{315732481}$

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

Class group and class number

$C_{15362}$, which has order $15362$ (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:  \( 161406.837641 \) (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

\(\Q(\sqrt{5}) \), 4.0.21125.1, 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
13Data not computed