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Properties

Label 20.10.2838818318...6944.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 3^{10}\cdot 71^{9}$
Root discriminant $23.59$
Ramified primes $2, 3, 71$
Class number $1$
Class group Trivial
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![1, 50, -6, -510, 514, 1470, -2883, 76, 3809, -2840, -1345, 2784, -826, -856, 727, -66, -143, 60, 2, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 2*x^18 + 60*x^17 - 143*x^16 - 66*x^15 + 727*x^14 - 856*x^13 - 826*x^12 + 2784*x^11 - 1345*x^10 - 2840*x^9 + 3809*x^8 + 76*x^7 - 2883*x^6 + 1470*x^5 + 514*x^4 - 510*x^3 - 6*x^2 + 50*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 2*x^18 + 60*x^17 - 143*x^16 - 66*x^15 + 727*x^14 - 856*x^13 - 826*x^12 + 2784*x^11 - 1345*x^10 - 2840*x^9 + 3809*x^8 + 76*x^7 - 2883*x^6 + 1470*x^5 + 514*x^4 - 510*x^3 - 6*x^2 + 50*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 2 x^{18} + 60 x^{17} - 143 x^{16} - 66 x^{15} + 727 x^{14} - 856 x^{13} - 826 x^{12} + 2784 x^{11} - 1345 x^{10} - 2840 x^{9} + 3809 x^{8} + 76 x^{7} - 2883 x^{6} + 1470 x^{5} + 514 x^{4} - 510 x^{3} - 6 x^{2} + 50 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2838818318108534328806866944=-\,2^{20}\cdot 3^{10}\cdot 71^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.59$
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:  $2, 3, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13} a^{17} - \frac{2}{13} a^{16} - \frac{6}{13} a^{15} + \frac{1}{13} a^{14} - \frac{4}{13} a^{13} - \frac{2}{13} a^{12} - \frac{5}{13} a^{11} + \frac{5}{13} a^{10} + \frac{5}{13} a^{9} + \frac{2}{13} a^{8} - \frac{4}{13} a^{7} - \frac{2}{13} a^{6} - \frac{5}{13} a^{4} + \frac{1}{13} a^{3} + \frac{5}{13} a^{2} - \frac{6}{13} a + \frac{3}{13}$, $\frac{1}{169} a^{18} + \frac{4}{169} a^{17} - \frac{70}{169} a^{16} - \frac{74}{169} a^{15} + \frac{28}{169} a^{14} + \frac{4}{13} a^{13} - \frac{30}{169} a^{12} - \frac{51}{169} a^{11} - \frac{4}{169} a^{10} + \frac{6}{169} a^{9} + \frac{8}{169} a^{8} - \frac{4}{13} a^{7} + \frac{27}{169} a^{6} - \frac{83}{169} a^{5} + \frac{23}{169} a^{4} + \frac{11}{169} a^{3} + \frac{76}{169} a^{2} + \frac{32}{169} a + \frac{83}{169}$, $\frac{1}{4282124315687469511} a^{19} - \frac{4853055111993683}{4282124315687469511} a^{18} + \frac{72797142046570387}{4282124315687469511} a^{17} - \frac{426203974377601905}{4282124315687469511} a^{16} + \frac{2115561978318465584}{4282124315687469511} a^{15} - \frac{462142659366970948}{4282124315687469511} a^{14} - \frac{1449272954599662625}{4282124315687469511} a^{13} + \frac{842069603195693514}{4282124315687469511} a^{12} - \frac{1719027098713658752}{4282124315687469511} a^{11} + \frac{734391309272604214}{4282124315687469511} a^{10} - \frac{405035062509835990}{4282124315687469511} a^{9} + \frac{240894029323159100}{4282124315687469511} a^{8} + \frac{396960302143128270}{4282124315687469511} a^{7} - \frac{1858624506827264065}{4282124315687469511} a^{6} + \frac{1722500188955076312}{4282124315687469511} a^{5} + \frac{925927220734278855}{4282124315687469511} a^{4} - \frac{58229423608369920}{329394178129805347} a^{3} - \frac{468314284944614555}{4282124315687469511} a^{2} - \frac{1311104836207493510}{4282124315687469511} a + \frac{555894690240807198}{4282124315687469511}$

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:  $14$
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:  \( 2724257.01428 \)
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

\(\Q(\sqrt{3}) \), 4.2.10224.2, 10.10.6323239406592.1

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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{10}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
2Data not computed
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$71$71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
71.10.9.7$x^{10} + 568$$10$$1$$9$$C_{10}$$[\ ]_{10}$