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Properties

Label 20.0.31490708085...9513.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 37^{3}\cdot 109^{3}\cdot 241^{5}$
Root discriminant $23.71$
Ramified primes $3, 37, 109, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21061, 7751, 82905, -27434, 84960, -94915, 50182, -41275, 29955, -13758, 6845, -870, -967, 683, -434, 160, -14, -10, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 13*x^18 - 10*x^17 - 14*x^16 + 160*x^15 - 434*x^14 + 683*x^13 - 967*x^12 - 870*x^11 + 6845*x^10 - 13758*x^9 + 29955*x^8 - 41275*x^7 + 50182*x^6 - 94915*x^5 + 84960*x^4 - 27434*x^3 + 82905*x^2 + 7751*x + 21061)
 
gp: K = bnfinit(x^20 - 6*x^19 + 13*x^18 - 10*x^17 - 14*x^16 + 160*x^15 - 434*x^14 + 683*x^13 - 967*x^12 - 870*x^11 + 6845*x^10 - 13758*x^9 + 29955*x^8 - 41275*x^7 + 50182*x^6 - 94915*x^5 + 84960*x^4 - 27434*x^3 + 82905*x^2 + 7751*x + 21061, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 13 x^{18} - 10 x^{17} - 14 x^{16} + 160 x^{15} - 434 x^{14} + 683 x^{13} - 967 x^{12} - 870 x^{11} + 6845 x^{10} - 13758 x^{9} + 29955 x^{8} - 41275 x^{7} + 50182 x^{6} - 94915 x^{5} + 84960 x^{4} - 27434 x^{3} + 82905 x^{2} + 7751 x + 21061 \)

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:  \(3149070808568813267726079513=3^{10}\cdot 37^{3}\cdot 109^{3}\cdot 241^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.71$
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:  $3, 37, 109, 241$
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}{1643033383696858576567594716627545437621962510853271} a^{19} + \frac{188171981514733280327719966746282033875244391581328}{1643033383696858576567594716627545437621962510853271} a^{18} - \frac{807072117223612273999937718456853918924416321416389}{1643033383696858576567594716627545437621962510853271} a^{17} - \frac{602480310134891366651677561488935419378338087419066}{1643033383696858576567594716627545437621962510853271} a^{16} + \frac{686348677241931810868039667717370625289924168718818}{1643033383696858576567594716627545437621962510853271} a^{15} - \frac{543353459184559307075629730865869096792935355967640}{1643033383696858576567594716627545437621962510853271} a^{14} - \frac{307947623138966338126750734169977362777680446587756}{1643033383696858576567594716627545437621962510853271} a^{13} + \frac{475160088024667479653340732776552647475977963252960}{1643033383696858576567594716627545437621962510853271} a^{12} - \frac{605178001915463898401355348970070162758049068533147}{1643033383696858576567594716627545437621962510853271} a^{11} - \frac{441049055092680210125564090796010525849281702567667}{1643033383696858576567594716627545437621962510853271} a^{10} + \frac{443265404165103269927754643788014241384116782423391}{1643033383696858576567594716627545437621962510853271} a^{9} - \frac{178589398996404135344911328041994339782885525778925}{1643033383696858576567594716627545437621962510853271} a^{8} + \frac{580465327441910272677009851189637447168652699725914}{1643033383696858576567594716627545437621962510853271} a^{7} - \frac{171573084931723025085596746168541358059179718704293}{1643033383696858576567594716627545437621962510853271} a^{6} + \frac{487640527843128571172702083634417150780673148302078}{1643033383696858576567594716627545437621962510853271} a^{5} + \frac{151200993434956937892249496319737411668855101094833}{1643033383696858576567594716627545437621962510853271} a^{4} + \frac{359922056921977751671849989139473573275998508063854}{1643033383696858576567594716627545437621962510853271} a^{3} + \frac{123537829233799454568318945849555240036788954073846}{1643033383696858576567594716627545437621962510853271} a^{2} + \frac{404438144844466630004317400572063535104473701800645}{1643033383696858576567594716627545437621962510853271} a + \frac{582712573460853736321653266464813145355137572986767}{1643033383696858576567594716627545437621962510853271}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{112922583919906254593880025}{271981066525018596466508713577} a^{19} - \frac{798178873005515567695549151}{271981066525018596466508713577} a^{18} + \frac{2176083833087870489561108141}{271981066525018596466508713577} a^{17} - \frac{2538581268659425161239513845}{271981066525018596466508713577} a^{16} - \frac{975199308546439804027667813}{271981066525018596466508713577} a^{15} + \frac{20783622607777098893178486305}{271981066525018596466508713577} a^{14} - \frac{68690535682323599723094363601}{271981066525018596466508713577} a^{13} + \frac{125744199688965346682821883932}{271981066525018596466508713577} a^{12} - \frac{173992158347911024007081449065}{271981066525018596466508713577} a^{11} - \frac{21295704868944664061545969636}{271981066525018596466508713577} a^{10} + \frac{937107375447560431358647996412}{271981066525018596466508713577} a^{9} - \frac{2421505965839789707811275022551}{271981066525018596466508713577} a^{8} + \frac{4866062348512245346003199093647}{271981066525018596466508713577} a^{7} - \frac{7583326301025061342870435890378}{271981066525018596466508713577} a^{6} + \frac{9243788586617417876743909402154}{271981066525018596466508713577} a^{5} - \frac{14258629033883655414073373510008}{271981066525018596466508713577} a^{4} + \frac{17755975969242629764334774155877}{271981066525018596466508713577} a^{3} - \frac{9018819202872480501595765431052}{271981066525018596466508713577} a^{2} + \frac{5965457261351704256914787107897}{271981066525018596466508713577} a - \frac{3664879367271032038402506784609}{271981066525018596466508713577} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 597728.148931 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1022 are not computed
Character table for t20n1022 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.0.236184579.1

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 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R $16{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $16{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ $16{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
37Data not computed
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.4.3.2$x^{4} - 3924$$4$$1$$3$$C_4$$[\ ]_{4}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
241Data not computed