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Properties

Label 20.0.80894366654...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{8}\cdot 3^{18}\cdot 5^{10}\cdot 17^{4}$
Root discriminant $13.98$
Ramified primes $2, 3, 5, 17$
Class number $1$
Class group Trivial
Galois group $D_5\wr C_2:C_2$ (as 20T96)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -36, 160, -468, 1008, -1710, 2406, -2916, 3108, -2934, 2469, -1878, 1314, -852, 510, -279, 138, -60, 22, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 22*x^18 - 60*x^17 + 138*x^16 - 279*x^15 + 510*x^14 - 852*x^13 + 1314*x^12 - 1878*x^11 + 2469*x^10 - 2934*x^9 + 3108*x^8 - 2916*x^7 + 2406*x^6 - 1710*x^5 + 1008*x^4 - 468*x^3 + 160*x^2 - 36*x + 4)
 
gp: K = bnfinit(x^20 - 6*x^19 + 22*x^18 - 60*x^17 + 138*x^16 - 279*x^15 + 510*x^14 - 852*x^13 + 1314*x^12 - 1878*x^11 + 2469*x^10 - 2934*x^9 + 3108*x^8 - 2916*x^7 + 2406*x^6 - 1710*x^5 + 1008*x^4 - 468*x^3 + 160*x^2 - 36*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 22 x^{18} - 60 x^{17} + 138 x^{16} - 279 x^{15} + 510 x^{14} - 852 x^{13} + 1314 x^{12} - 1878 x^{11} + 2469 x^{10} - 2934 x^{9} + 3108 x^{8} - 2916 x^{7} + 2406 x^{6} - 1710 x^{5} + 1008 x^{4} - 468 x^{3} + 160 x^{2} - 36 x + 4 \)

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:  \(80894366654422500000000=2^{8}\cdot 3^{18}\cdot 5^{10}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.98$
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:  $2, 3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{12} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{13} - \frac{1}{2} a^{8}$, $\frac{1}{3678098} a^{19} - \frac{188387}{1839049} a^{18} + \frac{97072}{1839049} a^{17} - \frac{455863}{1839049} a^{16} + \frac{287596}{1839049} a^{15} - \frac{405575}{3678098} a^{14} + \frac{550350}{1839049} a^{13} + \frac{344573}{1839049} a^{12} - \frac{93350}{1839049} a^{11} - \frac{520264}{1839049} a^{10} + \frac{224447}{3678098} a^{9} - \frac{649556}{1839049} a^{8} + \frac{470887}{1839049} a^{7} - \frac{258595}{1839049} a^{6} - \frac{654808}{1839049} a^{5} + \frac{437290}{1839049} a^{4} - \frac{156404}{1839049} a^{3} - \frac{625069}{1839049} a^{2} - \frac{778819}{1839049} a - \frac{903368}{1839049}$

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:  \( -\frac{48175592}{1839049} a^{19} + \frac{516486673}{3678098} a^{18} - \frac{1789058537}{3678098} a^{17} + \frac{4636496427}{3678098} a^{16} - \frac{10330746235}{3678098} a^{15} + \frac{10138299300}{1839049} a^{14} - \frac{36176075791}{3678098} a^{13} + \frac{58969837485}{3678098} a^{12} - \frac{88921941839}{3678098} a^{11} + \frac{124136777391}{3678098} a^{10} - \frac{79298068202}{1839049} a^{9} + \frac{181413579897}{3678098} a^{8} - \frac{183650233689}{3678098} a^{7} + \frac{163776893097}{3678098} a^{6} - \frac{127359034245}{3678098} a^{5} + \frac{41782840137}{1839049} a^{4} - \frac{21941136738}{1839049} a^{3} + \frac{8578920654}{1839049} a^{2} - \frac{2248799961}{1839049} a + \frac{303506770}{1839049} \) (order $6$)
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:  \( 5840.86639243 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2:C_2$ (as 20T96):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 16 conjugacy class representatives for $D_5\wr C_2:C_2$
Character table for $D_5\wr C_2:C_2$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 10.0.284419350000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
3Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$