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Properties

Label 20.0.47052523380...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{16}$
Root discriminant $30.45$
Ramified primes $2, 5, 11$
Class number $31$ (GRH)
Class group $[31]$ (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, 0, 45, 0, 470, 0, 2079, 0, 4588, 0, 5493, 0, 3694, 0, 1386, 0, 277, 0, 27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 27*x^18 + 277*x^16 + 1386*x^14 + 3694*x^12 + 5493*x^10 + 4588*x^8 + 2079*x^6 + 470*x^4 + 45*x^2 + 1)
 
gp: K = bnfinit(x^20 + 27*x^18 + 277*x^16 + 1386*x^14 + 3694*x^12 + 5493*x^10 + 4588*x^8 + 2079*x^6 + 470*x^4 + 45*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 27 x^{18} + 277 x^{16} + 1386 x^{14} + 3694 x^{12} + 5493 x^{10} + 4588 x^{8} + 2079 x^{6} + 470 x^{4} + 45 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(470525233802978928640000000000=2^{20}\cdot 5^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.45$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(220=2^{2}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(69,·)$, $\chi_{220}(71,·)$, $\chi_{220}(9,·)$, $\chi_{220}(141,·)$, $\chi_{220}(81,·)$, $\chi_{220}(89,·)$, $\chi_{220}(201,·)$, $\chi_{220}(91,·)$, $\chi_{220}(159,·)$, $\chi_{220}(31,·)$, $\chi_{220}(169,·)$, $\chi_{220}(199,·)$, $\chi_{220}(111,·)$, $\chi_{220}(49,·)$, $\chi_{220}(179,·)$, $\chi_{220}(181,·)$, $\chi_{220}(119,·)$, $\chi_{220}(59,·)$, $\chi_{220}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{41507731} a^{18} + \frac{2426979}{41507731} a^{16} + \frac{6970730}{41507731} a^{14} - \frac{10869172}{41507731} a^{12} - \frac{7022661}{41507731} a^{10} - \frac{5696945}{41507731} a^{8} + \frac{13139048}{41507731} a^{6} - \frac{18931934}{41507731} a^{4} + \frac{4730290}{41507731} a^{2} + \frac{20043876}{41507731}$, $\frac{1}{41507731} a^{19} + \frac{2426979}{41507731} a^{17} + \frac{6970730}{41507731} a^{15} - \frac{10869172}{41507731} a^{13} - \frac{7022661}{41507731} a^{11} - \frac{5696945}{41507731} a^{9} + \frac{13139048}{41507731} a^{7} - \frac{18931934}{41507731} a^{5} + \frac{4730290}{41507731} a^{3} + \frac{20043876}{41507731} a$

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

Class group and class number

$C_{31}$, which has order $31$ (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:  \( -\frac{14370}{125401} a^{19} - \frac{416120}{125401} a^{17} - \frac{4714345}{125401} a^{15} - \frac{27040100}{125401} a^{13} - \frac{85587256}{125401} a^{11} - \frac{153298999}{125401} a^{9} - \frac{152848944}{125401} a^{7} - \frac{79313708}{125401} a^{5} - \frac{18211390}{125401} a^{3} - \frac{1204854}{125401} a \) (order $4$)
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:  \( 140644.599182 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.219503494144.1, 10.0.685948419200000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$