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Properties

Label 20.20.3774226305...5625.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{10}\cdot 17^{10}\cdot 61^{8}$
Root discriminant $47.74$
Ramified primes $5, 17, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times D_5$ (as 20T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-389, -2541, 12343, 79483, 89047, -122728, -251687, 40097, 258755, 32874, -138737, -30367, 42920, 9770, -7831, -1499, 819, 109, -45, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 45*x^18 + 109*x^17 + 819*x^16 - 1499*x^15 - 7831*x^14 + 9770*x^13 + 42920*x^12 - 30367*x^11 - 138737*x^10 + 32874*x^9 + 258755*x^8 + 40097*x^7 - 251687*x^6 - 122728*x^5 + 89047*x^4 + 79483*x^3 + 12343*x^2 - 2541*x - 389)
 
gp: K = bnfinit(x^20 - 3*x^19 - 45*x^18 + 109*x^17 + 819*x^16 - 1499*x^15 - 7831*x^14 + 9770*x^13 + 42920*x^12 - 30367*x^11 - 138737*x^10 + 32874*x^9 + 258755*x^8 + 40097*x^7 - 251687*x^6 - 122728*x^5 + 89047*x^4 + 79483*x^3 + 12343*x^2 - 2541*x - 389, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 45 x^{18} + 109 x^{17} + 819 x^{16} - 1499 x^{15} - 7831 x^{14} + 9770 x^{13} + 42920 x^{12} - 30367 x^{11} - 138737 x^{10} + 32874 x^{9} + 258755 x^{8} + 40097 x^{7} - 251687 x^{6} - 122728 x^{5} + 89047 x^{4} + 79483 x^{3} + 12343 x^{2} - 2541 x - 389 \)

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:  \(3774226305410017540250773134765625=5^{10}\cdot 17^{10}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.74$
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:  $5, 17, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{35} a^{16} + \frac{1}{35} a^{15} + \frac{12}{35} a^{14} + \frac{11}{35} a^{13} + \frac{2}{5} a^{12} + \frac{16}{35} a^{11} - \frac{6}{35} a^{10} - \frac{3}{35} a^{9} + \frac{17}{35} a^{8} + \frac{12}{35} a^{6} + \frac{1}{5} a^{5} + \frac{12}{35} a^{4} + \frac{13}{35} a^{3} - \frac{4}{35} a^{2} + \frac{2}{7} a + \frac{16}{35}$, $\frac{1}{35} a^{17} + \frac{11}{35} a^{15} - \frac{1}{35} a^{14} + \frac{3}{35} a^{13} + \frac{2}{35} a^{12} + \frac{13}{35} a^{11} + \frac{3}{35} a^{10} - \frac{3}{7} a^{9} - \frac{17}{35} a^{8} + \frac{12}{35} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{35} a^{4} - \frac{17}{35} a^{3} + \frac{2}{5} a^{2} + \frac{6}{35} a - \frac{16}{35}$, $\frac{1}{436345} a^{18} - \frac{5947}{436345} a^{17} + \frac{5256}{436345} a^{16} + \frac{192597}{436345} a^{15} - \frac{27123}{87269} a^{14} + \frac{61311}{436345} a^{13} - \frac{55701}{436345} a^{12} - \frac{124673}{436345} a^{11} - \frac{89716}{436345} a^{10} + \frac{214118}{436345} a^{9} - \frac{181949}{436345} a^{8} - \frac{91844}{436345} a^{7} - \frac{3291}{12467} a^{6} - \frac{175124}{436345} a^{5} - \frac{90899}{436345} a^{4} + \frac{59783}{436345} a^{3} + \frac{24148}{436345} a^{2} + \frac{214642}{436345} a + \frac{146242}{436345}$, $\frac{1}{639455793069085268082653665} a^{19} - \frac{8767046285544693918}{13050118225899699348625585} a^{18} + \frac{2785808047097381409364503}{639455793069085268082653665} a^{17} - \frac{8322564523088895717379929}{639455793069085268082653665} a^{16} + \frac{147424159431424202909571331}{639455793069085268082653665} a^{15} - \frac{135851867258615843148734988}{639455793069085268082653665} a^{14} + \frac{177132561643236192520024929}{639455793069085268082653665} a^{13} + \frac{130159882688797791803843782}{639455793069085268082653665} a^{12} - \frac{168252417179050239217110826}{639455793069085268082653665} a^{11} - \frac{3325787750230484487650906}{18270165516259579088075819} a^{10} - \frac{94451108580101103158498601}{639455793069085268082653665} a^{9} + \frac{58542486054344765547121472}{127891158613817053616530733} a^{8} + \frac{22580875198744033778037044}{639455793069085268082653665} a^{7} + \frac{123948336182220175092505894}{639455793069085268082653665} a^{6} + \frac{217229636751807007793386079}{639455793069085268082653665} a^{5} + \frac{110339515251006896102312933}{639455793069085268082653665} a^{4} - \frac{143854355727594936755757269}{639455793069085268082653665} a^{3} - \frac{15779088047833988192844588}{91350827581297895440379095} a^{2} - \frac{295963765715929519903061981}{639455793069085268082653665} a + \frac{119534383986528043749604142}{639455793069085268082653665}$

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:  \( 22717345132.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_5$ (as 20T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 16 conjugacy class representatives for $C_2^2\times D_5$
Character table for $C_2^2\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}, \sqrt{17})\), 5.5.26884225.1, 10.10.12286946415460625.1, 10.10.3613807769253125.1, 10.10.61434732077303125.1

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

Sibling fields

Galois closure: data not computed
Degree 20 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$