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Properties

Label 20.4.41153942158...0000.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{24}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $42.73$
Ramified primes $2, 5, 3469$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-20975, -107400, 314885, -293560, 110721, 800, -84758, 65854, -35209, 28024, -21587, 16524, -10739, 5322, -2050, 714, -191, 18, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 13*x^18 + 18*x^17 - 191*x^16 + 714*x^15 - 2050*x^14 + 5322*x^13 - 10739*x^12 + 16524*x^11 - 21587*x^10 + 28024*x^9 - 35209*x^8 + 65854*x^7 - 84758*x^6 + 800*x^5 + 110721*x^4 - 293560*x^3 + 314885*x^2 - 107400*x - 20975)
 
gp: K = bnfinit(x^20 - 6*x^19 + 13*x^18 + 18*x^17 - 191*x^16 + 714*x^15 - 2050*x^14 + 5322*x^13 - 10739*x^12 + 16524*x^11 - 21587*x^10 + 28024*x^9 - 35209*x^8 + 65854*x^7 - 84758*x^6 + 800*x^5 + 110721*x^4 - 293560*x^3 + 314885*x^2 - 107400*x - 20975, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 13 x^{18} + 18 x^{17} - 191 x^{16} + 714 x^{15} - 2050 x^{14} + 5322 x^{13} - 10739 x^{12} + 16524 x^{11} - 21587 x^{10} + 28024 x^{9} - 35209 x^{8} + 65854 x^{7} - 84758 x^{6} + 800 x^{5} + 110721 x^{4} - 293560 x^{3} + 314885 x^{2} - 107400 x - 20975 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(411539421581712055500800000000000=2^{24}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.73$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{15} + \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{15} + \frac{1}{5} a^{14} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{235151180587442370009917412075626248623941825311797695} a^{19} - \frac{19560196850154224957483605147894939632403134293366413}{235151180587442370009917412075626248623941825311797695} a^{18} + \frac{13841613324725418529030888937826779893025749480943051}{235151180587442370009917412075626248623941825311797695} a^{17} - \frac{1995857211601833209807120961912623628834878724986422}{47030236117488474001983482415125249724788365062359539} a^{16} - \frac{68887754158335173653522645423266316006841033067804771}{235151180587442370009917412075626248623941825311797695} a^{15} + \frac{62067035752274757531395256346815569151444684310873152}{235151180587442370009917412075626248623941825311797695} a^{14} - \frac{15213655728119514405033926977070342395293962990658885}{47030236117488474001983482415125249724788365062359539} a^{13} - \frac{86402339185620597424869593634404944962065184826792231}{235151180587442370009917412075626248623941825311797695} a^{12} - \frac{109920824212333693750592319306474466985903692897695621}{235151180587442370009917412075626248623941825311797695} a^{11} + \frac{72714560400356303237048732159311794298035616544855908}{235151180587442370009917412075626248623941825311797695} a^{10} - \frac{22514589427481348865481549977386819231131043933359102}{235151180587442370009917412075626248623941825311797695} a^{9} - \frac{45951833083755485302868660640089365432688591135451382}{235151180587442370009917412075626248623941825311797695} a^{8} - \frac{20591950516793974562516924656539469986340439357998322}{235151180587442370009917412075626248623941825311797695} a^{7} + \frac{117344148317683201297167199070213172400884756865183778}{235151180587442370009917412075626248623941825311797695} a^{6} + \frac{10054232408017615673270313539241569604025351829218031}{235151180587442370009917412075626248623941825311797695} a^{5} - \frac{1477032224322508770839369712176119821261695646240134}{3854937386679383114916678886485676206949865988717995} a^{4} + \frac{67155838484360583763688604928622997195990919121250123}{235151180587442370009917412075626248623941825311797695} a^{3} - \frac{79533477404556024537162332184546940166365586420002669}{235151180587442370009917412075626248623941825311797695} a^{2} + \frac{3565719801307883031779044777410070999788300898709470}{47030236117488474001983482415125249724788365062359539} a - \frac{13412820357538101705357162072097859559931006432084837}{47030236117488474001983482415125249724788365062359539}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 112279593.614 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed