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Properties

Label 20.4.13743895347...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{52}\cdot 5^{15}$
Root discriminant $20.27$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -5, 10, -35, 120, -232, 216, -135, 290, -681, 954, -849, 468, -110, -60, 79, -46, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 19*x^18 - 46*x^17 + 79*x^16 - 60*x^15 - 110*x^14 + 468*x^13 - 849*x^12 + 954*x^11 - 681*x^10 + 290*x^9 - 135*x^8 + 216*x^7 - 232*x^6 + 120*x^5 - 35*x^4 + 10*x^3 - 5*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 19*x^18 - 46*x^17 + 79*x^16 - 60*x^15 - 110*x^14 + 468*x^13 - 849*x^12 + 954*x^11 - 681*x^10 + 290*x^9 - 135*x^8 + 216*x^7 - 232*x^6 + 120*x^5 - 35*x^4 + 10*x^3 - 5*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 19 x^{18} - 46 x^{17} + 79 x^{16} - 60 x^{15} - 110 x^{14} + 468 x^{13} - 849 x^{12} + 954 x^{11} - 681 x^{10} + 290 x^{9} - 135 x^{8} + 216 x^{7} - 232 x^{6} + 120 x^{5} - 35 x^{4} + 10 x^{3} - 5 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(137438953472000000000000000=2^{52}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.27$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{8} a^{18} + \frac{1}{8} a^{17} + \frac{1}{8} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{3}{8} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{1274233704349912} a^{19} - \frac{64126294239547}{1274233704349912} a^{18} + \frac{239501277172581}{1274233704349912} a^{17} - \frac{59381656843441}{318558426087478} a^{16} - \frac{174647001768717}{637116852174956} a^{15} - \frac{214812838790493}{637116852174956} a^{14} - \frac{301777254823337}{637116852174956} a^{13} - \frac{26833268945268}{159279213043739} a^{12} + \frac{178777976181145}{1274233704349912} a^{11} + \frac{563945502578605}{1274233704349912} a^{10} - \frac{153765610158871}{1274233704349912} a^{9} + \frac{59077195915088}{159279213043739} a^{8} + \frac{27129922984363}{159279213043739} a^{7} - \frac{105818644920411}{318558426087478} a^{6} - \frac{126985656819687}{318558426087478} a^{5} - \frac{30357567893243}{159279213043739} a^{4} + \frac{263656832897673}{1274233704349912} a^{3} - \frac{38606647073283}{1274233704349912} a^{2} - \frac{183655809629411}{1274233704349912} a - \frac{59964954352133}{318558426087478}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 246251.671322 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 5.1.256000.1, 10.2.327680000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
5Data not computed