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Properties

Label 20.0.36976851779...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{42}\cdot 3^{16}\cdot 5^{9}$
Root discriminant $21.30$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $S_5$ (as 20T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20, -56, -24, 368, -638, -56, 1620, -2224, 849, 968, -1410, 612, 203, -396, 184, 20, -61, 24, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 2*x^18 + 24*x^17 - 61*x^16 + 20*x^15 + 184*x^14 - 396*x^13 + 203*x^12 + 612*x^11 - 1410*x^10 + 968*x^9 + 849*x^8 - 2224*x^7 + 1620*x^6 - 56*x^5 - 638*x^4 + 368*x^3 - 24*x^2 - 56*x + 20)
 
gp: K = bnfinit(x^20 - 4*x^19 + 2*x^18 + 24*x^17 - 61*x^16 + 20*x^15 + 184*x^14 - 396*x^13 + 203*x^12 + 612*x^11 - 1410*x^10 + 968*x^9 + 849*x^8 - 2224*x^7 + 1620*x^6 - 56*x^5 - 638*x^4 + 368*x^3 - 24*x^2 - 56*x + 20, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 2 x^{18} + 24 x^{17} - 61 x^{16} + 20 x^{15} + 184 x^{14} - 396 x^{13} + 203 x^{12} + 612 x^{11} - 1410 x^{10} + 968 x^{9} + 849 x^{8} - 2224 x^{7} + 1620 x^{6} - 56 x^{5} - 638 x^{4} + 368 x^{3} - 24 x^{2} - 56 x + 20 \)

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:  \(369768517790072832000000000=2^{42}\cdot 3^{16}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.30$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{4771595030967382642} a^{19} - \frac{196055538580180625}{4771595030967382642} a^{18} - \frac{288841710267702037}{2385797515483691321} a^{17} + \frac{597004105571181475}{4771595030967382642} a^{16} - \frac{2110115335440931167}{4771595030967382642} a^{15} + \frac{2381504258700775283}{4771595030967382642} a^{14} + \frac{1012769272826438833}{2385797515483691321} a^{13} - \frac{173593148773485001}{4771595030967382642} a^{12} - \frac{2010540763481361559}{4771595030967382642} a^{11} + \frac{1640544735196725093}{4771595030967382642} a^{10} + \frac{1016221232736121629}{2385797515483691321} a^{9} - \frac{129638876375609325}{4771595030967382642} a^{8} + \frac{1487040212831096211}{4771595030967382642} a^{7} - \frac{2164844415769319787}{4771595030967382642} a^{6} + \frac{1027322600993439717}{2385797515483691321} a^{5} - \frac{1219674804241082187}{4771595030967382642} a^{4} - \frac{688480579989283040}{2385797515483691321} a^{3} - \frac{375353983254114439}{2385797515483691321} a^{2} - \frac{213676825607445200}{2385797515483691321} a - \frac{913005361515047637}{2385797515483691321}$

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:  $9$
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:  \( 383250.383654 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 20T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

10.2.214990848000.1

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

Sibling fields

Degree 5 sibling: 5.1.103680.1
Degree 6 sibling: 6.2.10368000.1
Degree 10 siblings: 10.2.214990848000.1, 10.2.1343692800000.3
Degree 12 sibling: 12.4.107495424000000.1
Degree 15 sibling: 15.3.2229025112064000000.1
Degree 20 siblings: 20.4.28888165452349440000000000.1, Deg 20
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.8.16.9$x^{8} + 2 x^{4} + 8 x + 12$$4$$2$$16$$S_4$$[8/3, 8/3]_{3}^{2}$
2.12.26.73$x^{12} - 2 x^{8} + 4 x^{7} + 4 x^{6} + 2 x^{4} + 4 x^{3} + 4 x^{2} - 2$$12$$1$$26$$S_4$$[8/3, 8/3]_{3}^{2}$
3Data not computed
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$