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Properties

Label 22.2.10265350264...0625.1
Degree $22$
Signature $[2, 10]$
Discriminant $5^{16}\cdot 11^{20}$
Root discriminant $28.51$
Ramified primes $5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{11}:D_{11}.C_2$ (as 22T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -10, 155, 325, -451, -3936, -9954, -15445, -16556, -12496, -6129, -1095, 1084, 974, 36, -440, -241, 39, 71, 5, -11, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 11*x^20 + 5*x^19 + 71*x^18 + 39*x^17 - 241*x^16 - 440*x^15 + 36*x^14 + 974*x^13 + 1084*x^12 - 1095*x^11 - 6129*x^10 - 12496*x^9 - 16556*x^8 - 15445*x^7 - 9954*x^6 - 3936*x^5 - 451*x^4 + 325*x^3 + 155*x^2 - 10*x - 5)
 
gp: K = bnfinit(x^22 - x^21 - 11*x^20 + 5*x^19 + 71*x^18 + 39*x^17 - 241*x^16 - 440*x^15 + 36*x^14 + 974*x^13 + 1084*x^12 - 1095*x^11 - 6129*x^10 - 12496*x^9 - 16556*x^8 - 15445*x^7 - 9954*x^6 - 3936*x^5 - 451*x^4 + 325*x^3 + 155*x^2 - 10*x - 5, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} - 11 x^{20} + 5 x^{19} + 71 x^{18} + 39 x^{17} - 241 x^{16} - 440 x^{15} + 36 x^{14} + 974 x^{13} + 1084 x^{12} - 1095 x^{11} - 6129 x^{10} - 12496 x^{9} - 16556 x^{8} - 15445 x^{7} - 9954 x^{6} - 3936 x^{5} - 451 x^{4} + 325 x^{3} + 155 x^{2} - 10 x - 5 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(102653502644738770935211181640625=5^{16}\cdot 11^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.51$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{15} + \frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6}$, $\frac{1}{12} a^{18} + \frac{1}{12} a^{16} - \frac{1}{4} a^{15} + \frac{1}{12} a^{14} - \frac{1}{12} a^{13} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{4} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{12} a + \frac{1}{4}$, $\frac{1}{228} a^{19} - \frac{1}{76} a^{18} - \frac{3}{76} a^{17} - \frac{5}{38} a^{16} - \frac{3}{38} a^{15} + \frac{5}{57} a^{14} - \frac{7}{228} a^{13} + \frac{3}{38} a^{12} - \frac{14}{57} a^{11} + \frac{21}{76} a^{10} - \frac{13}{114} a^{9} - \frac{11}{228} a^{8} + \frac{43}{114} a^{7} + \frac{43}{114} a^{6} - \frac{55}{228} a^{5} - \frac{103}{228} a^{4} + \frac{1}{38} a^{3} - \frac{113}{228} a^{2} - \frac{15}{38} a + \frac{73}{228}$, $\frac{1}{1140} a^{20} + \frac{1}{57} a^{18} - \frac{1}{12} a^{17} - \frac{46}{285} a^{16} + \frac{9}{38} a^{15} + \frac{41}{228} a^{14} + \frac{7}{228} a^{13} + \frac{3}{95} a^{12} + \frac{17}{228} a^{11} + \frac{25}{228} a^{10} - \frac{11}{76} a^{9} - \frac{289}{1140} a^{8} - \frac{53}{114} a^{7} - \frac{43}{228} a^{6} - \frac{23}{114} a^{5} - \frac{379}{1140} a^{4} + \frac{1}{4} a^{3} - \frac{25}{228} a^{2} - \frac{47}{228} a + \frac{97}{228}$, $\frac{1}{168095357221495513399080} a^{21} - \frac{2482733978762729377}{84047678610747756699540} a^{20} - \frac{9138474360710865319}{33619071444299102679816} a^{19} - \frac{318888930418499955215}{8404767861074775669954} a^{18} - \frac{3803676555901632113823}{56031785740498504466360} a^{17} + \frac{8528342898792281962283}{84047678610747756699540} a^{16} + \frac{1966297472053948161773}{33619071444299102679816} a^{15} + \frac{2150603521098042351601}{11206357148099700893272} a^{14} - \frac{19404208353280869068009}{168095357221495513399080} a^{13} + \frac{38093563759701223302481}{168095357221495513399080} a^{12} - \frac{1061617929775781779211}{11206357148099700893272} a^{11} + \frac{2093977614467398388446}{4202383930537387834977} a^{10} - \frac{18524283305940935538259}{168095357221495513399080} a^{9} - \frac{12974914834057145053973}{56031785740498504466360} a^{8} + \frac{1391783462700924672891}{11206357148099700893272} a^{7} - \frac{6354438187091246859175}{16809535722149551339908} a^{6} - \frac{22246131974908806675257}{84047678610747756699540} a^{5} + \frac{13921058502830335875581}{28015892870249252233180} a^{4} - \frac{2308323720235659562941}{11206357148099700893272} a^{3} + \frac{5995230184371862382975}{16809535722149551339908} a^{2} - \frac{15363662735553717408329}{33619071444299102679816} a - \frac{15790145597457941348071}{33619071444299102679816}$

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:  \( 67615839.0404 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{11}:D_{11}.C_2$ (as 22T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 484
The 34 conjugacy class representatives for $C_{11}:D_{11}.C_2$
Character table for $C_{11}:D_{11}.C_2$ is not computed

Intermediate fields

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

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

Sibling fields

Degree 22 siblings: data not computed
Degree 44 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$11.11.20.1$x^{11} - 11 x^{10} + 374$$11$$1$$20$$C_{11}$$[2]$
11.11.0.1$x^{11} - x + 3$$1$$11$$0$$C_{11}$$[\ ]^{11}$