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Properties

Label 20.4.32473988377...0464.3
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{17}\cdot 89^{4}$
Root discriminant $26.64$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43, 278, 662, 194, -1660, -1220, 2365, 1637, -3277, 1099, -550, 760, -124, -79, -29, 21, -1, 11, -9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 9*x^18 + 11*x^17 - x^16 + 21*x^15 - 29*x^14 - 79*x^13 - 124*x^12 + 760*x^11 - 550*x^10 + 1099*x^9 - 3277*x^8 + 1637*x^7 + 2365*x^6 - 1220*x^5 - 1660*x^4 + 194*x^3 + 662*x^2 + 278*x + 43)
 
gp: K = bnfinit(x^20 - x^19 - 9*x^18 + 11*x^17 - x^16 + 21*x^15 - 29*x^14 - 79*x^13 - 124*x^12 + 760*x^11 - 550*x^10 + 1099*x^9 - 3277*x^8 + 1637*x^7 + 2365*x^6 - 1220*x^5 - 1660*x^4 + 194*x^3 + 662*x^2 + 278*x + 43, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 9 x^{18} + 11 x^{17} - x^{16} + 21 x^{15} - 29 x^{14} - 79 x^{13} - 124 x^{12} + 760 x^{11} - 550 x^{10} + 1099 x^{9} - 3277 x^{8} + 1637 x^{7} + 2365 x^{6} - 1220 x^{5} - 1660 x^{4} + 194 x^{3} + 662 x^{2} + 278 x + 43 \)

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:  \(32473988377432635504517950464=2^{10}\cdot 11^{17}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.64$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{4}{11} a^{14} + \frac{1}{11} a^{13} - \frac{3}{11} a^{12} - \frac{2}{11} a^{11} - \frac{1}{11} a^{10} - \frac{1}{11} a^{9} - \frac{1}{11} a^{8} - \frac{1}{11} a^{7} - \frac{1}{11} a^{6} - \frac{1}{11} a^{5} - \frac{2}{11} a^{4} - \frac{5}{11} a^{3} - \frac{2}{11} a^{2} + \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} - \frac{4}{11} a^{14} + \frac{4}{11} a^{13} - \frac{1}{11} a^{12} - \frac{4}{11} a^{11} + \frac{3}{11} a^{10} + \frac{3}{11} a^{9} + \frac{3}{11} a^{8} + \frac{3}{11} a^{7} + \frac{3}{11} a^{6} + \frac{2}{11} a^{5} + \frac{3}{11} a^{4} - \frac{4}{11} a^{3} - \frac{1}{11} a^{2} + \frac{4}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{17} - \frac{2}{11} a^{14} + \frac{3}{11} a^{13} - \frac{5}{11} a^{12} - \frac{5}{11} a^{11} - \frac{1}{11} a^{10} - \frac{1}{11} a^{9} - \frac{1}{11} a^{8} - \frac{1}{11} a^{7} - \frac{2}{11} a^{6} - \frac{1}{11} a^{5} - \frac{1}{11} a^{4} + \frac{1}{11} a^{3} - \frac{4}{11} a^{2} + \frac{4}{11} a + \frac{4}{11}$, $\frac{1}{11} a^{18} - \frac{3}{11} a^{13} - \frac{5}{11} a^{11} - \frac{3}{11} a^{10} - \frac{3}{11} a^{9} - \frac{3}{11} a^{8} - \frac{4}{11} a^{7} - \frac{3}{11} a^{6} - \frac{3}{11} a^{5} - \frac{3}{11} a^{4} - \frac{3}{11} a^{3} - \frac{3}{11} a + \frac{2}{11}$, $\frac{1}{2621468171357609112158004295097} a^{19} + \frac{406079186683050518861380518}{2621468171357609112158004295097} a^{18} + \frac{49679386143106001543764669069}{2621468171357609112158004295097} a^{17} - \frac{36136017243287652906563916616}{2621468171357609112158004295097} a^{16} + \frac{22526861237346394629011288820}{2621468171357609112158004295097} a^{15} - \frac{755061144640705754943155646655}{2621468171357609112158004295097} a^{14} - \frac{863705432704297074873836986458}{2621468171357609112158004295097} a^{13} + \frac{1295611586935465269924161397977}{2621468171357609112158004295097} a^{12} + \frac{1178375018291221722166282224675}{2621468171357609112158004295097} a^{11} + \frac{472652792126413468772306744798}{2621468171357609112158004295097} a^{10} - \frac{939191869534277799184752676984}{2621468171357609112158004295097} a^{9} + \frac{570564532265055643812640318354}{2621468171357609112158004295097} a^{8} - \frac{1031362236485521097305239644965}{2621468171357609112158004295097} a^{7} + \frac{867650535197759097565838937585}{2621468171357609112158004295097} a^{6} - \frac{597150058051254257777358354501}{2621468171357609112158004295097} a^{5} - \frac{986387700872325088255428905068}{2621468171357609112158004295097} a^{4} + \frac{62132104342789858801591928045}{238315288305237192014364026827} a^{3} + \frac{287193534934077292093422101803}{2621468171357609112158004295097} a^{2} - \frac{1125542736940358615871572260315}{2621468171357609112158004295097} a - \frac{152254176466883769766553287899}{2621468171357609112158004295097}$

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:  $11$
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:  \( 2126408.67142 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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 $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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}$$[2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$