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Properties

Label 20.20.2570887044...0625.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{10}\cdot 460181^{2}\cdot 1114969^{2}$
Root discriminant $33.15$
Ramified primes $5, 460181, 1114969$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1021

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -54, -84, 534, 818, -2259, -2917, 4712, 5134, -5200, -4870, 3127, 2502, -1043, -662, 203, 84, -22, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 22*x^18 + 84*x^17 + 203*x^16 - 662*x^15 - 1043*x^14 + 2502*x^13 + 3127*x^12 - 4870*x^11 - 5200*x^10 + 5134*x^9 + 4712*x^8 - 2917*x^7 - 2259*x^6 + 818*x^5 + 534*x^4 - 84*x^3 - 54*x^2 + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 22*x^18 + 84*x^17 + 203*x^16 - 662*x^15 - 1043*x^14 + 2502*x^13 + 3127*x^12 - 4870*x^11 - 5200*x^10 + 5134*x^9 + 4712*x^8 - 2917*x^7 - 2259*x^6 + 818*x^5 + 534*x^4 - 84*x^3 - 54*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 22 x^{18} + 84 x^{17} + 203 x^{16} - 662 x^{15} - 1043 x^{14} + 2502 x^{13} + 3127 x^{12} - 4870 x^{11} - 5200 x^{10} + 5134 x^{9} + 4712 x^{8} - 2917 x^{7} - 2259 x^{6} + 818 x^{5} + 534 x^{4} - 84 x^{3} - 54 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2570887044316499162825400390625=5^{10}\cdot 460181^{2}\cdot 1114969^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.15$
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:  $5, 460181, 1114969$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{7681690435576651} a^{19} + \frac{1038919877981103}{7681690435576651} a^{18} + \frac{867896752389221}{7681690435576651} a^{17} + \frac{1675609549454775}{7681690435576651} a^{16} + \frac{1870040953065483}{7681690435576651} a^{15} - \frac{722316646168829}{7681690435576651} a^{14} - \frac{2551349681895249}{7681690435576651} a^{13} - \frac{3005250181065911}{7681690435576651} a^{12} + \frac{1732355282303281}{7681690435576651} a^{11} + \frac{31117295648169}{590899264275127} a^{10} + \frac{121711591929803}{590899264275127} a^{9} + \frac{3229123599921660}{7681690435576651} a^{8} + \frac{1707152171883029}{7681690435576651} a^{7} - \frac{108854092559150}{590899264275127} a^{6} - \frac{1058334926368015}{7681690435576651} a^{5} + \frac{1397835326487913}{7681690435576651} a^{4} + \frac{2980534647000163}{7681690435576651} a^{3} + \frac{850022759133247}{7681690435576651} a^{2} + \frac{2499642671413174}{7681690435576651} a + \frac{2013253706414787}{7681690435576651}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $19$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 451418457.714 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1021:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7257600
The 84 conjugacy class representatives for t20n1021 are not computed
Character table for t20n1021 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.513087549389.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $18{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
460181Data not computed
1114969Data not computed