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Properties

Label 20.0.41115038262...2272.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{42}\cdot 2657^{7}$
Root discriminant $67.72$
Ramified primes $2, 2657$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group 20T547

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![90850706, -276218960, 473508600, -543999108, 495962812, -378201868, 247514958, -138934976, 69078496, -29303884, 11379146, -3705440, 1180861, -301748, 80556, -15820, 3536, -508, 92, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 92*x^18 - 508*x^17 + 3536*x^16 - 15820*x^15 + 80556*x^14 - 301748*x^13 + 1180861*x^12 - 3705440*x^11 + 11379146*x^10 - 29303884*x^9 + 69078496*x^8 - 138934976*x^7 + 247514958*x^6 - 378201868*x^5 + 495962812*x^4 - 543999108*x^3 + 473508600*x^2 - 276218960*x + 90850706)
 
gp: K = bnfinit(x^20 - 8*x^19 + 92*x^18 - 508*x^17 + 3536*x^16 - 15820*x^15 + 80556*x^14 - 301748*x^13 + 1180861*x^12 - 3705440*x^11 + 11379146*x^10 - 29303884*x^9 + 69078496*x^8 - 138934976*x^7 + 247514958*x^6 - 378201868*x^5 + 495962812*x^4 - 543999108*x^3 + 473508600*x^2 - 276218960*x + 90850706, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 92 x^{18} - 508 x^{17} + 3536 x^{16} - 15820 x^{15} + 80556 x^{14} - 301748 x^{13} + 1180861 x^{12} - 3705440 x^{11} + 11379146 x^{10} - 29303884 x^{9} + 69078496 x^{8} - 138934976 x^{7} + 247514958 x^{6} - 378201868 x^{5} + 495962812 x^{4} - 543999108 x^{3} + 473508600 x^{2} - 276218960 x + 90850706 \)

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:  \(4111503826253885814528363168920502272=2^{42}\cdot 2657^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.72$
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:  $2, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{34} a^{16} + \frac{6}{17} a^{15} + \frac{6}{17} a^{13} - \frac{5}{34} a^{12} + \frac{6}{17} a^{11} - \frac{3}{17} a^{10} - \frac{4}{17} a^{9} - \frac{8}{17} a^{7} - \frac{5}{17} a^{6} + \frac{4}{17} a^{5} + \frac{5}{17} a^{4} - \frac{2}{17} a^{3} - \frac{7}{17} a - \frac{2}{17}$, $\frac{1}{34} a^{17} - \frac{4}{17} a^{15} + \frac{6}{17} a^{14} - \frac{13}{34} a^{13} + \frac{2}{17} a^{12} - \frac{7}{17} a^{11} - \frac{2}{17} a^{10} - \frac{3}{17} a^{9} - \frac{8}{17} a^{8} + \frac{6}{17} a^{7} - \frac{4}{17} a^{6} + \frac{8}{17} a^{5} + \frac{6}{17} a^{4} + \frac{7}{17} a^{3} - \frac{7}{17} a^{2} - \frac{3}{17} a + \frac{7}{17}$, $\frac{1}{34} a^{18} + \frac{3}{17} a^{15} - \frac{13}{34} a^{14} - \frac{1}{17} a^{13} + \frac{7}{17} a^{12} - \frac{5}{17} a^{11} + \frac{7}{17} a^{10} - \frac{6}{17} a^{9} + \frac{6}{17} a^{8} + \frac{2}{17} a^{6} + \frac{4}{17} a^{5} - \frac{4}{17} a^{4} - \frac{6}{17} a^{3} - \frac{3}{17} a^{2} + \frac{2}{17} a + \frac{1}{17}$, $\frac{1}{34172011567813186842736292618577184999499528547790890462447100906633021146926} a^{19} - \frac{340561755267773164254152884843759672303296605578987310251830811607098025665}{34172011567813186842736292618577184999499528547790890462447100906633021146926} a^{18} - \frac{234734820522029153523718995729447025931293374081978074955923888928706669553}{17086005783906593421368146309288592499749764273895445231223550453316510573463} a^{17} - \frac{2085046147585103716139637617053502469748646323818745735445854230911856397}{2010118327518422755455076036386893235264678149870052380143947112154883596878} a^{16} - \frac{596010790504474777078673299328912860190432771975918008242846108631701771249}{2010118327518422755455076036386893235264678149870052380143947112154883596878} a^{15} + \frac{14856154775342118554037609052150171287582709497211067393097017481598992816491}{34172011567813186842736292618577184999499528547790890462447100906633021146926} a^{14} + \frac{1591438799371315498854869750432106593271158580793254112848232046804268325464}{17086005783906593421368146309288592499749764273895445231223550453316510573463} a^{13} - \frac{2304457111618042016966170195781979643240012420443402671296579210579735129661}{34172011567813186842736292618577184999499528547790890462447100906633021146926} a^{12} - \frac{6259851226323058066513843520972502207279866637459431843007812737126860331396}{17086005783906593421368146309288592499749764273895445231223550453316510573463} a^{11} - \frac{3686620079693729934655423394613913251928031796749403988303249620858120898044}{17086005783906593421368146309288592499749764273895445231223550453316510573463} a^{10} - \frac{1095184287712475276179956033952606305958182599312473915880043201293395330831}{17086005783906593421368146309288592499749764273895445231223550453316510573463} a^{9} + \frac{490566992730614660936603868948239309292358900029494113247904037269316390660}{1005059163759211377727538018193446617632339074935026190071973556077441798439} a^{8} + \frac{6064281412786951843679761835718174338071223785125228036236434106804647739817}{17086005783906593421368146309288592499749764273895445231223550453316510573463} a^{7} + \frac{87133294381515312037272608174483767075209195299473721954959586233514552508}{899263462310873337966744542594136447355250751257655012169660550174553188077} a^{6} - \frac{4058305175804633940956398696448745042833992942479195384872812749857142163760}{17086005783906593421368146309288592499749764273895445231223550453316510573463} a^{5} + \frac{3831484056143262353857089845501469498330405648963006927305905969655568527645}{17086005783906593421368146309288592499749764273895445231223550453316510573463} a^{4} - \frac{3037545733257615831342150899975848278039815550585739057433490411534997377886}{17086005783906593421368146309288592499749764273895445231223550453316510573463} a^{3} + \frac{7231430949815640042906531227583255531210596342532928228664561161888040771216}{17086005783906593421368146309288592499749764273895445231223550453316510573463} a^{2} - \frac{120487784877594432256345214623700470469883302434540219418079886397130870732}{1005059163759211377727538018193446617632339074935026190071973556077441798439} a - \frac{8087065301718519483420230935302353190769014396748118037885153949935534849424}{17086005783906593421368146309288592499749764273895445231223550453316510573463}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( -1 \) (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:  \( 1374798594.16 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 28800
The 41 conjugacy class representatives for t20n547
Character table for t20n547 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.680192.2, 10.6.925322313728.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 24 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 $20$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.12.26.64$x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{6} - 2 x^{4} + 4 x^{3} + 2$$12$$1$$26$$S_3 \times C_2^2$$[2, 3]_{3}^{2}$
2657Data not computed