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Properties

Label 20.20.7869545944...3125.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{10}\cdot 29^{5}\cdot 211^{8}$
Root discriminant $44.14$
Ramified primes $5, 29, 211$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![179, 528, -3045, -6600, 16403, 21890, -40081, -32819, 51642, 26337, -37808, -12098, 16223, 3233, -4062, -487, 566, 37, -39, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 39*x^18 + 37*x^17 + 566*x^16 - 487*x^15 - 4062*x^14 + 3233*x^13 + 16223*x^12 - 12098*x^11 - 37808*x^10 + 26337*x^9 + 51642*x^8 - 32819*x^7 - 40081*x^6 + 21890*x^5 + 16403*x^4 - 6600*x^3 - 3045*x^2 + 528*x + 179)
 
gp: K = bnfinit(x^20 - x^19 - 39*x^18 + 37*x^17 + 566*x^16 - 487*x^15 - 4062*x^14 + 3233*x^13 + 16223*x^12 - 12098*x^11 - 37808*x^10 + 26337*x^9 + 51642*x^8 - 32819*x^7 - 40081*x^6 + 21890*x^5 + 16403*x^4 - 6600*x^3 - 3045*x^2 + 528*x + 179, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 39 x^{18} + 37 x^{17} + 566 x^{16} - 487 x^{15} - 4062 x^{14} + 3233 x^{13} + 16223 x^{12} - 12098 x^{11} - 37808 x^{10} + 26337 x^{9} + 51642 x^{8} - 32819 x^{7} - 40081 x^{6} + 21890 x^{5} + 16403 x^{4} - 6600 x^{3} - 3045 x^{2} + 528 x + 179 \)

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:  \(786954594434420875688580751953125=5^{10}\cdot 29^{5}\cdot 211^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.14$
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:  $5, 29, 211$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{467648490052607314380401} a^{19} + \frac{107230205104993676335069}{467648490052607314380401} a^{18} + \frac{214521439010218586808066}{467648490052607314380401} a^{17} - \frac{85045531640544319457090}{467648490052607314380401} a^{16} + \frac{196996065108692232128503}{467648490052607314380401} a^{15} - \frac{126733067828436866312126}{467648490052607314380401} a^{14} + \frac{32737005264634368038723}{467648490052607314380401} a^{13} + \frac{216124360363040423933355}{467648490052607314380401} a^{12} + \frac{81687966478725916326001}{467648490052607314380401} a^{11} - \frac{21860369584390847116186}{467648490052607314380401} a^{10} - \frac{186359606374416829751947}{467648490052607314380401} a^{9} - \frac{168203572456852768457792}{467648490052607314380401} a^{8} + \frac{130489328106152555206695}{467648490052607314380401} a^{7} + \frac{24451455517805085025236}{467648490052607314380401} a^{6} - \frac{47978653422912707167529}{467648490052607314380401} a^{5} + \frac{147040885504697388703628}{467648490052607314380401} a^{4} - \frac{117810906550182664216962}{467648490052607314380401} a^{3} - \frac{108363154805755442087673}{467648490052607314380401} a^{2} - \frac{189766542418341975482964}{467648490052607314380401} a - \frac{115895596655568137618276}{467648490052607314380401}$

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:  \( 9457791448.16 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 10.10.6194123253125.1

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

Sibling fields

Degree 20 siblings: 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 $20$ $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
5Data not computed
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.10.5.2$x^{10} - 707281 x^{2} + 225622639$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
211Data not computed