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Properties

Label 20.4.16384000000...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{36}\cdot 5^{22}$
Root discriminant $20.45$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $S_5$ (as 20T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25, 100, -75, -50, 25, -20, 90, -140, 55, 60, 11, -230, 385, -400, 350, -292, 215, -120, 45, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 45*x^18 - 120*x^17 + 215*x^16 - 292*x^15 + 350*x^14 - 400*x^13 + 385*x^12 - 230*x^11 + 11*x^10 + 60*x^9 + 55*x^8 - 140*x^7 + 90*x^6 - 20*x^5 + 25*x^4 - 50*x^3 - 75*x^2 + 100*x - 25)
 
gp: K = bnfinit(x^20 - 10*x^19 + 45*x^18 - 120*x^17 + 215*x^16 - 292*x^15 + 350*x^14 - 400*x^13 + 385*x^12 - 230*x^11 + 11*x^10 + 60*x^9 + 55*x^8 - 140*x^7 + 90*x^6 - 20*x^5 + 25*x^4 - 50*x^3 - 75*x^2 + 100*x - 25, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 45 x^{18} - 120 x^{17} + 215 x^{16} - 292 x^{15} + 350 x^{14} - 400 x^{13} + 385 x^{12} - 230 x^{11} + 11 x^{10} + 60 x^{9} + 55 x^{8} - 140 x^{7} + 90 x^{6} - 20 x^{5} + 25 x^{4} - 50 x^{3} - 75 x^{2} + 100 x - 25 \)

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:  \(163840000000000000000000000=2^{36}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.45$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{6}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{10} + \frac{2}{5} a^{7} - \frac{1}{2} a^{6} - \frac{2}{5} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{11} + \frac{2}{5} a^{8} - \frac{1}{2} a^{7} - \frac{2}{5} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{10} + \frac{2}{5} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{2}{5} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{10} a^{15} - \frac{1}{10} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{2}{5} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a$, $\frac{1}{10} a^{16} - \frac{1}{2} a^{8} + \frac{2}{5} a^{6} - \frac{1}{2}$, $\frac{1}{10} a^{17} - \frac{1}{2} a^{9} + \frac{2}{5} a^{7} - \frac{1}{2} a$, $\frac{1}{61850} a^{18} - \frac{9}{61850} a^{17} - \frac{442}{30925} a^{16} + \frac{1091}{61850} a^{15} - \frac{917}{30925} a^{14} + \frac{2309}{61850} a^{13} + \frac{717}{30925} a^{12} + \frac{2197}{30925} a^{11} - \frac{3281}{61850} a^{10} + \frac{2592}{30925} a^{9} - \frac{15}{1237} a^{8} - \frac{1989}{6185} a^{7} - \frac{80}{1237} a^{6} - \frac{765}{2474} a^{5} - \frac{674}{6185} a^{4} + \frac{1161}{2474} a^{3} + \frac{557}{2474} a^{2} + \frac{1203}{2474} a + \frac{439}{1237}$, $\frac{1}{78611350} a^{19} + \frac{313}{39305675} a^{18} + \frac{2838501}{78611350} a^{17} - \frac{1154009}{78611350} a^{16} + \frac{1705291}{78611350} a^{15} - \frac{2807491}{78611350} a^{14} - \frac{1823673}{39305675} a^{13} - \frac{1206273}{39305675} a^{12} + \frac{3808697}{39305675} a^{11} - \frac{5480001}{78611350} a^{10} - \frac{765937}{1572227} a^{9} + \frac{3835803}{7861135} a^{8} + \frac{3946391}{15722270} a^{7} + \frac{936715}{3144454} a^{6} - \frac{6455421}{15722270} a^{5} - \frac{1038409}{3144454} a^{4} - \frac{639}{2474} a^{3} - \frac{395282}{1572227} a^{2} - \frac{202091}{1572227} a - \frac{214798}{1572227}$

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:  \( 331332.875442 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 20T32):

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

\(\Q(\sqrt{5}) \), 10.2.12800000000000.1

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

Sibling fields

Degree 5 sibling: 5.1.800000.1
Degree 6 sibling: 6.2.3200000.1
Degree 10 siblings: 10.2.12800000000000.1, 10.2.3200000000000.2
Degree 12 sibling: 12.4.256000000000000.1
Degree 15 sibling: Deg 15
Degree 20 siblings: Deg 20, 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.8.16.8$x^{8} + 8 x^{5} + 12$$4$$2$$16$$S_4$$[8/3, 8/3]_{3}^{2}$
2.12.20.37$x^{12} - 6 x^{10} - x^{8} + 4 x^{6} + 3 x^{4} + 2 x^{2} - 7$$6$$2$$20$$S_4$$[8/3, 8/3]_{3}^{2}$
$5$5.10.11.1$x^{10} + 20 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.1$x^{10} + 20 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$