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Properties

Label 20.4.90782435907...8125.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{10}\cdot 5^{17}\cdot 67^{4}$
Root discriminant $15.77$
Ramified primes $3, 5, 67$
Class number $1$
Class group Trivial
Galois group $C_4\times S_5$ (as 20T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 3, -26, -27, 81, 47, -173, 10, 209, -130, -101, 205, -142, 49, -29, 42, -39, 22, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 22*x^18 - 39*x^17 + 42*x^16 - 29*x^15 + 49*x^14 - 142*x^13 + 205*x^12 - 101*x^11 - 130*x^10 + 209*x^9 + 10*x^8 - 173*x^7 + 47*x^6 + 81*x^5 - 27*x^4 - 26*x^3 + 3*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 22*x^18 - 39*x^17 + 42*x^16 - 29*x^15 + 49*x^14 - 142*x^13 + 205*x^12 - 101*x^11 - 130*x^10 + 209*x^9 + 10*x^8 - 173*x^7 + 47*x^6 + 81*x^5 - 27*x^4 - 26*x^3 + 3*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 22 x^{18} - 39 x^{17} + 42 x^{16} - 29 x^{15} + 49 x^{14} - 142 x^{13} + 205 x^{12} - 101 x^{11} - 130 x^{10} + 209 x^{9} + 10 x^{8} - 173 x^{7} + 47 x^{6} + 81 x^{5} - 27 x^{4} - 26 x^{3} + 3 x^{2} + 5 x + 1 \)

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:  \(907824359076690673828125=3^{10}\cdot 5^{17}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.77$
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:  $3, 5, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{85020733097939} a^{19} - \frac{36333138191853}{85020733097939} a^{18} + \frac{25643644805758}{85020733097939} a^{17} - \frac{30525586466036}{85020733097939} a^{16} + \frac{21986697831261}{85020733097939} a^{15} + \frac{24586386368631}{85020733097939} a^{14} + \frac{39185305328724}{85020733097939} a^{13} - \frac{21363628795259}{85020733097939} a^{12} + \frac{16215647433114}{85020733097939} a^{11} - \frac{22165760551261}{85020733097939} a^{10} - \frac{2798220435446}{85020733097939} a^{9} - \frac{26671781216382}{85020733097939} a^{8} + \frac{281067614743}{85020733097939} a^{7} + \frac{35819363027762}{85020733097939} a^{6} - \frac{7969245495166}{85020733097939} a^{5} - \frac{20580862886928}{85020733097939} a^{4} + \frac{17484605765382}{85020733097939} a^{3} - \frac{23358362573380}{85020733097939} a^{2} + \frac{10536658875311}{85020733097939} a + \frac{11765364947104}{85020733097939}$

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:  \( 10142.1455616 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times S_5$ (as 20T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 28 conjugacy class representatives for $C_4\times S_5$
Character table for $C_4\times S_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 5.1.5025.1, 10.2.3156328125.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 24 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 $20$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
5Data not computed
$67$67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.8.4.1$x^{8} + 17956 x^{4} - 300763 x^{2} + 80604484$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$