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Properties

Label 20.0.20050185375...4277.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{17}\cdot 1583^{3}$
Root discriminant $23.18$
Ramified primes $11, 1583$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T432

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 11, 44, -187, 330, -496, 860, -1230, 985, -97, -595, 551, -67, -208, 164, -21, -24, 14, 1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + x^18 + 14*x^17 - 24*x^16 - 21*x^15 + 164*x^14 - 208*x^13 - 67*x^12 + 551*x^11 - 595*x^10 - 97*x^9 + 985*x^8 - 1230*x^7 + 860*x^6 - 496*x^5 + 330*x^4 - 187*x^3 + 44*x^2 + 11*x + 11)
 
gp: K = bnfinit(x^20 - 4*x^19 + x^18 + 14*x^17 - 24*x^16 - 21*x^15 + 164*x^14 - 208*x^13 - 67*x^12 + 551*x^11 - 595*x^10 - 97*x^9 + 985*x^8 - 1230*x^7 + 860*x^6 - 496*x^5 + 330*x^4 - 187*x^3 + 44*x^2 + 11*x + 11, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + x^{18} + 14 x^{17} - 24 x^{16} - 21 x^{15} + 164 x^{14} - 208 x^{13} - 67 x^{12} + 551 x^{11} - 595 x^{10} - 97 x^{9} + 985 x^{8} - 1230 x^{7} + 860 x^{6} - 496 x^{5} + 330 x^{4} - 187 x^{3} + 44 x^{2} + 11 x + 11 \)

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:  \(2005018537548922694563074277=11^{17}\cdot 1583^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.18$
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:  $11, 1583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} - \frac{5}{11} a^{17} + \frac{3}{11} a^{16} + \frac{4}{11} a^{15} - \frac{4}{11} a^{14} + \frac{4}{11} a^{13} - \frac{4}{11} a^{12} + \frac{4}{11} a^{11} - \frac{4}{11} a^{10} + \frac{4}{11} a^{9} - \frac{4}{11} a^{8} + \frac{5}{11} a^{7} + \frac{2}{11} a^{6} - \frac{4}{11} a^{5}$, $\frac{1}{3789619394185409888750383} a^{19} + \frac{61274359202123102728416}{3789619394185409888750383} a^{18} + \frac{920520313364261204497352}{3789619394185409888750383} a^{17} - \frac{727718456576892788562927}{3789619394185409888750383} a^{16} + \frac{172580682430459397528750}{3789619394185409888750383} a^{15} + \frac{454189723527316307101200}{3789619394185409888750383} a^{14} - \frac{586713160160786698450623}{3789619394185409888750383} a^{13} - \frac{541241037266365359049310}{3789619394185409888750383} a^{12} + \frac{1695210722401613290728280}{3789619394185409888750383} a^{11} + \frac{1198708000974477466793664}{3789619394185409888750383} a^{10} - \frac{306304896316757939096048}{3789619394185409888750383} a^{9} - \frac{759382316321879698968560}{3789619394185409888750383} a^{8} - \frac{1345635570192980906920613}{3789619394185409888750383} a^{7} + \frac{927402574771164181222381}{3789619394185409888750383} a^{6} + \frac{1271118605088243160543403}{3789619394185409888750383} a^{5} - \frac{154391549244293215864098}{344510854016855444431853} a^{4} - \frac{108694841703109660267730}{344510854016855444431853} a^{3} + \frac{120618876547850807468101}{344510854016855444431853} a^{2} - \frac{109024209619580205352766}{344510854016855444431853} a + \frac{24417874783815092892145}{344510854016855444431853}$

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:  \( -1 \) (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:  \( 222104.62467 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T432:

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 t20n432 are not computed
Character table for t20n432 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.0.339330108623.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
$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}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1583Data not computed