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Properties

Label 20.20.5120338176...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{8}\cdot 5^{26}\cdot 41^{10}$
Root discriminant $68.46$
Ramified primes $2, 5, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5\wr C_2$ (as 20T55)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1636, 10440, -20400, -101220, 180400, 144522, -327140, -86400, 273020, 26400, -128051, -4320, 36400, 360, -6400, -12, 680, 0, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 40*x^18 + 680*x^16 - 12*x^15 - 6400*x^14 + 360*x^13 + 36400*x^12 - 4320*x^11 - 128051*x^10 + 26400*x^9 + 273020*x^8 - 86400*x^7 - 327140*x^6 + 144522*x^5 + 180400*x^4 - 101220*x^3 - 20400*x^2 + 10440*x + 1636)
 
gp: K = bnfinit(x^20 - 40*x^18 + 680*x^16 - 12*x^15 - 6400*x^14 + 360*x^13 + 36400*x^12 - 4320*x^11 - 128051*x^10 + 26400*x^9 + 273020*x^8 - 86400*x^7 - 327140*x^6 + 144522*x^5 + 180400*x^4 - 101220*x^3 - 20400*x^2 + 10440*x + 1636, 1)
 

Normalized defining polynomial

\( x^{20} - 40 x^{18} + 680 x^{16} - 12 x^{15} - 6400 x^{14} + 360 x^{13} + 36400 x^{12} - 4320 x^{11} - 128051 x^{10} + 26400 x^{9} + 273020 x^{8} - 86400 x^{7} - 327140 x^{6} + 144522 x^{5} + 180400 x^{4} - 101220 x^{3} - 20400 x^{2} + 10440 x + 1636 \)

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:  \(5120338176785431289672851562500000000=2^{8}\cdot 5^{26}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.46$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{5} + \frac{2}{5}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{6} + \frac{2}{5} a$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{7} + \frac{2}{5} a^{2}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{8} + \frac{2}{5} a^{3}$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{9} + \frac{2}{5} a^{4}$, $\frac{1}{100} a^{15} + \frac{1}{100} a^{10} - \frac{1}{2} a^{8} - \frac{19}{50} a^{5} + \frac{7}{25}$, $\frac{1}{1300} a^{16} - \frac{3}{65} a^{14} + \frac{3}{130} a^{12} + \frac{1}{1300} a^{11} + \frac{21}{130} a^{9} + \frac{4}{13} a^{8} - \frac{23}{130} a^{7} + \frac{231}{650} a^{6} + \frac{18}{65} a^{4} - \frac{2}{13} a^{3} - \frac{29}{65} a^{2} - \frac{43}{325} a + \frac{5}{13}$, $\frac{1}{1300} a^{17} + \frac{1}{260} a^{15} + \frac{3}{130} a^{13} + \frac{1}{1300} a^{12} + \frac{3}{260} a^{10} + \frac{4}{13} a^{9} + \frac{21}{65} a^{8} + \frac{231}{650} a^{7} - \frac{11}{26} a^{5} - \frac{2}{13} a^{4} - \frac{29}{65} a^{3} - \frac{43}{325} a^{2} + \frac{5}{13} a - \frac{2}{5}$, $\frac{1}{6500} a^{18} + \frac{1}{6500} a^{17} + \frac{1}{3250} a^{16} + \frac{9}{3250} a^{15} - \frac{1}{130} a^{14} + \frac{161}{6500} a^{13} - \frac{89}{6500} a^{12} + \frac{68}{1625} a^{11} + \frac{149}{3250} a^{10} + \frac{61}{130} a^{9} + \frac{101}{3250} a^{8} + \frac{613}{1625} a^{7} - \frac{224}{1625} a^{6} - \frac{557}{3250} a^{5} + \frac{23}{65} a^{4} + \frac{742}{1625} a^{3} - \frac{133}{1625} a^{2} + \frac{384}{1625} a - \frac{544}{1625}$, $\frac{1}{6500} a^{19} + \frac{1}{6500} a^{17} + \frac{1}{6500} a^{16} - \frac{3}{6500} a^{15} - \frac{189}{6500} a^{14} - \frac{1}{26} a^{13} - \frac{89}{6500} a^{12} + \frac{11}{6500} a^{11} + \frac{217}{6500} a^{10} + \frac{901}{3250} a^{9} - \frac{1}{13} a^{8} + \frac{51}{3250} a^{7} - \frac{162}{1625} a^{6} - \frac{739}{1625} a^{5} + \frac{59}{125} a^{4} - \frac{1}{13} a^{3} - \frac{558}{1625} a^{2} - \frac{283}{1625} a - \frac{226}{1625}$

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:  \( 2258149291720 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2$ (as 20T55):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{205}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{41})\), 10.10.452563285156250000.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.15.2$x^{10} - 15 x^{6} + 5$$10$$1$$15$$F_5$$[7/4]_{4}$
41Data not computed