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Properties

Label 20.0.73301084864...6816.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 31^{18}$
Root discriminant $43.98$
Ramified primes $2, 31$
Class number $123$ (GRH)
Class group $[123]$ (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![625, 0, -2750, 0, 5879, 0, -5372, 0, 1587, 0, 441, 0, 15, 0, 184, 0, 18, 0, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^18 + 18*x^16 + 184*x^14 + 15*x^12 + 441*x^10 + 1587*x^8 - 5372*x^6 + 5879*x^4 - 2750*x^2 + 625)
 
gp: K = bnfinit(x^20 - 3*x^18 + 18*x^16 + 184*x^14 + 15*x^12 + 441*x^10 + 1587*x^8 - 5372*x^6 + 5879*x^4 - 2750*x^2 + 625, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{18} + 18 x^{16} + 184 x^{14} + 15 x^{12} + 441 x^{10} + 1587 x^{8} - 5372 x^{6} + 5879 x^{4} - 2750 x^{2} + 625 \)

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:  \(733010848644118650331248974626816=2^{20}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.98$
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:  $2, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(124=2^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{124}(1,·)$, $\chi_{124}(33,·)$, $\chi_{124}(77,·)$, $\chi_{124}(15,·)$, $\chi_{124}(85,·)$, $\chi_{124}(23,·)$, $\chi_{124}(89,·)$, $\chi_{124}(27,·)$, $\chi_{124}(29,·)$, $\chi_{124}(95,·)$, $\chi_{124}(97,·)$, $\chi_{124}(35,·)$, $\chi_{124}(101,·)$, $\chi_{124}(39,·)$, $\chi_{124}(109,·)$, $\chi_{124}(47,·)$, $\chi_{124}(91,·)$, $\chi_{124}(123,·)$, $\chi_{124}(61,·)$, $\chi_{124}(63,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} - \frac{1}{5} a$, $\frac{1}{25} a^{10} - \frac{2}{25} a^{6} + \frac{1}{25} a^{2}$, $\frac{1}{25} a^{11} - \frac{2}{25} a^{7} + \frac{1}{25} a^{3}$, $\frac{1}{25} a^{12} - \frac{2}{25} a^{8} + \frac{1}{25} a^{4}$, $\frac{1}{25} a^{13} - \frac{2}{25} a^{9} + \frac{1}{25} a^{5}$, $\frac{1}{25} a^{14} + \frac{2}{25} a^{6} - \frac{3}{25} a^{2}$, $\frac{1}{125} a^{15} + \frac{2}{125} a^{11} - \frac{7}{125} a^{7} + \frac{4}{125} a^{3}$, $\frac{1}{125} a^{16} + \frac{2}{125} a^{12} - \frac{7}{125} a^{8} + \frac{4}{125} a^{4}$, $\frac{1}{125} a^{17} + \frac{2}{125} a^{13} - \frac{7}{125} a^{9} + \frac{4}{125} a^{5}$, $\frac{1}{20835115625} a^{18} - \frac{3393894}{20835115625} a^{16} + \frac{20979447}{20835115625} a^{14} - \frac{32221218}{20835115625} a^{12} + \frac{113909878}{20835115625} a^{10} - \frac{472523732}{20835115625} a^{8} + \frac{1965044}{39990625} a^{6} - \frac{9893315531}{20835115625} a^{4} + \frac{51644577}{166680925} a^{2} - \frac{4965984}{33336185}$, $\frac{1}{20835115625} a^{19} - \frac{3393894}{20835115625} a^{17} + \frac{20979447}{20835115625} a^{15} - \frac{32221218}{20835115625} a^{13} + \frac{113909878}{20835115625} a^{11} - \frac{472523732}{20835115625} a^{9} + \frac{1965044}{39990625} a^{7} - \frac{1559269281}{20835115625} a^{5} + \frac{51644577}{166680925} a^{3} + \frac{15035727}{33336185} a$

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

Class group and class number

$C_{123}$, which has order $123$ (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:  \( \frac{156623}{62194375} a^{19} - \frac{468377}{62194375} a^{17} + \frac{2746601}{62194375} a^{15} + \frac{28968506}{62194375} a^{13} + \frac{1558634}{62194375} a^{11} + \frac{55332944}{62194375} a^{9} + \frac{444947}{119375} a^{7} - \frac{885938323}{62194375} a^{5} + \frac{154222026}{12438875} a^{3} - \frac{1701188}{497555} a \) (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:  \( 95285261.3293 \) (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

\(\Q(\sqrt{-31}) \), \(\Q(\sqrt{31}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{31})\), 5.5.923521.1, 10.0.26439622160671.1, 10.10.27074173092527104.1, 10.0.873360422339584.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
31Data not computed