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Properties

Label 20.0.23027255619...6960.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5\cdot 61^{4}\cdot 397^{4}\cdot 1277$
Root discriminant $23.34$
Ramified primes $2, 5, 61, 397, 1277$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 20T887

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 10, 73, 220, -122, -744, 1470, -1068, -383, 1752, -1977, 1088, 97, -858, 1008, -758, 419, -174, 52, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 52*x^18 - 174*x^17 + 419*x^16 - 758*x^15 + 1008*x^14 - 858*x^13 + 97*x^12 + 1088*x^11 - 1977*x^10 + 1752*x^9 - 383*x^8 - 1068*x^7 + 1470*x^6 - 744*x^5 - 122*x^4 + 220*x^3 + 73*x^2 + 10*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 52*x^18 - 174*x^17 + 419*x^16 - 758*x^15 + 1008*x^14 - 858*x^13 + 97*x^12 + 1088*x^11 - 1977*x^10 + 1752*x^9 - 383*x^8 - 1068*x^7 + 1470*x^6 - 744*x^5 - 122*x^4 + 220*x^3 + 73*x^2 + 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 52 x^{18} - 174 x^{17} + 419 x^{16} - 758 x^{15} + 1008 x^{14} - 858 x^{13} + 97 x^{12} + 1088 x^{11} - 1977 x^{10} + 1752 x^{9} - 383 x^{8} - 1068 x^{7} + 1470 x^{6} - 744 x^{5} - 122 x^{4} + 220 x^{3} + 73 x^{2} + 10 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2302725561923832607497256960=2^{20}\cdot 5\cdot 61^{4}\cdot 397^{4}\cdot 1277\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.34$
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:  $2, 5, 61, 397, 1277$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{181787683781648284981661} a^{19} - \frac{36240479985433722396247}{181787683781648284981661} a^{18} + \frac{76148573083619792196948}{181787683781648284981661} a^{17} - \frac{44957773212727977634259}{181787683781648284981661} a^{16} + \frac{48112782675713563543372}{181787683781648284981661} a^{15} + \frac{87536928955639856660019}{181787683781648284981661} a^{14} - \frac{22022629445555476229094}{181787683781648284981661} a^{13} + \frac{88601129474153466922684}{181787683781648284981661} a^{12} - \frac{28342015946188404748195}{181787683781648284981661} a^{11} + \frac{24521571945059330819691}{181787683781648284981661} a^{10} + \frac{84196295100667532483848}{181787683781648284981661} a^{9} + \frac{52782422034067337281679}{181787683781648284981661} a^{8} + \frac{54242974516338898436319}{181787683781648284981661} a^{7} - \frac{18265377128713461509706}{181787683781648284981661} a^{6} - \frac{31438350978154506203109}{181787683781648284981661} a^{5} + \frac{74745567120665132278997}{181787683781648284981661} a^{4} - \frac{82113662561041243099993}{181787683781648284981661} a^{3} - \frac{87988134419282423032223}{181787683781648284981661} a^{2} - \frac{88072845568064768259181}{181787683781648284981661} a + \frac{59517280023082720584372}{181787683781648284981661}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{418154033652475}{1498264024696429} a^{19} - \frac{4248841060403626}{1498264024696429} a^{18} + \frac{22428841746737886}{1498264024696429} a^{17} - \frac{76385711766326228}{1498264024696429} a^{16} + \frac{187626650151798306}{1498264024696429} a^{15} - \frac{347710144169682657}{1498264024696429} a^{14} + \frac{479099625393174489}{1498264024696429} a^{13} - \frac{439394622176230613}{1498264024696429} a^{12} + \frac{116948754673569486}{1498264024696429} a^{11} + \frac{429659638441921428}{1498264024696429} a^{10} - \frac{891932646224064096}{1498264024696429} a^{9} + \frac{878687748811053781}{1498264024696429} a^{8} - \frac{311727184922956842}{1498264024696429} a^{7} - \frac{382725296817308771}{1498264024696429} a^{6} + \frac{667542644259897201}{1498264024696429} a^{5} - \frac{419819304160825170}{1498264024696429} a^{4} + \frac{24953463208764987}{1498264024696429} a^{3} + \frac{79877068737766636}{1498264024696429} a^{2} + \frac{21036095525368731}{1498264024696429} a + \frac{2271647223816144}{1498264024696429} \) (order $4$)
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:  \( 195751.092016 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T887:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 245760
The 201 conjugacy class representatives for t20n887 are not computed
Character table for t20n887 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.24217.1, 10.0.600538203136.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
61Data not computed
397Data not computed
1277Data not computed