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Properties

Label 20.20.3525175555...1856.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 3^{10}\cdot 11^{18}$
Root discriminant $42.40$
Ramified primes $2, 3, 11$
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![1, -20, -292, -450, 3516, 6736, -16067, -26542, 33501, 40262, -33034, -25808, 16726, 7644, -4374, -1104, 590, 76, -39, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 39*x^18 + 76*x^17 + 590*x^16 - 1104*x^15 - 4374*x^14 + 7644*x^13 + 16726*x^12 - 25808*x^11 - 33034*x^10 + 40262*x^9 + 33501*x^8 - 26542*x^7 - 16067*x^6 + 6736*x^5 + 3516*x^4 - 450*x^3 - 292*x^2 - 20*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 39*x^18 + 76*x^17 + 590*x^16 - 1104*x^15 - 4374*x^14 + 7644*x^13 + 16726*x^12 - 25808*x^11 - 33034*x^10 + 40262*x^9 + 33501*x^8 - 26542*x^7 - 16067*x^6 + 6736*x^5 + 3516*x^4 - 450*x^3 - 292*x^2 - 20*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 39 x^{18} + 76 x^{17} + 590 x^{16} - 1104 x^{15} - 4374 x^{14} + 7644 x^{13} + 16726 x^{12} - 25808 x^{11} - 33034 x^{10} + 40262 x^{9} + 33501 x^{8} - 26542 x^{7} - 16067 x^{6} + 6736 x^{5} + 3516 x^{4} - 450 x^{3} - 292 x^{2} - 20 x + 1 \)

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:  \(352517555563337816067682238201856=2^{30}\cdot 3^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.40$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(264=2^{3}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(197,·)$, $\chi_{264}(161,·)$, $\chi_{264}(17,·)$, $\chi_{264}(37,·)$, $\chi_{264}(149,·)$, $\chi_{264}(25,·)$, $\chi_{264}(101,·)$, $\chi_{264}(157,·)$, $\chi_{264}(133,·)$, $\chi_{264}(97,·)$, $\chi_{264}(229,·)$, $\chi_{264}(65,·)$, $\chi_{264}(41,·)$, $\chi_{264}(173,·)$, $\chi_{264}(29,·)$, $\chi_{264}(49,·)$, $\chi_{264}(181,·)$, $\chi_{264}(233,·)$, $\chi_{264}(169,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{241439363232784676036897281177} a^{19} + \frac{110217171731609918564159947199}{241439363232784676036897281177} a^{18} - \frac{66294939526375959495988363849}{241439363232784676036897281177} a^{17} - \frac{72006901491115173520222714164}{241439363232784676036897281177} a^{16} - \frac{91189850769176465663512669974}{241439363232784676036897281177} a^{15} + \frac{89164734695257262645788370601}{241439363232784676036897281177} a^{14} - \frac{91438449960733721428853497689}{241439363232784676036897281177} a^{13} + \frac{48384837447938497254542806953}{241439363232784676036897281177} a^{12} - \frac{115662238875054095821983913077}{241439363232784676036897281177} a^{11} + \frac{111306754155727704416855344039}{241439363232784676036897281177} a^{10} + \frac{43691275926753962694814358938}{241439363232784676036897281177} a^{9} - \frac{109537213643200824775459829691}{241439363232784676036897281177} a^{8} - \frac{24825391593781383936635729}{241439363232784676036897281177} a^{7} - \frac{73597345008969059922258275912}{241439363232784676036897281177} a^{6} + \frac{120461965046469615062901389947}{241439363232784676036897281177} a^{5} + \frac{119222907306568569736470936323}{241439363232784676036897281177} a^{4} + \frac{71571845191020112188201156373}{241439363232784676036897281177} a^{3} + \frac{99312039309002808894891247596}{241439363232784676036897281177} a^{2} + \frac{47198583902223375093925935100}{241439363232784676036897281177} a + \frac{92869382213135204308298558664}{241439363232784676036897281177}$

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:  \( 4400553067.16 \) (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

\(\Q(\sqrt{66}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 10.10.18775450875101184.1, \(\Q(\zeta_{33})^+\), 10.10.7024111812608.1

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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
3Data not computed
11Data not computed