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Properties

Label 22.0.21324856255...6691.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{20}\cdot 11^{19}$
Root discriminant $21.53$
Ramified primes $3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_{11}$ (as 22T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 18, -9, -36, -36, -99, 15, 372, 168, -426, -198, 193, 44, -2, 12, -51, 48, -33, 24, -12, 7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 + 7*x^20 - 12*x^19 + 24*x^18 - 33*x^17 + 48*x^16 - 51*x^15 + 12*x^14 - 2*x^13 + 44*x^12 + 193*x^11 - 198*x^10 - 426*x^9 + 168*x^8 + 372*x^7 + 15*x^6 - 99*x^5 - 36*x^4 - 36*x^3 - 9*x^2 + 18*x + 9)
 
gp: K = bnfinit(x^22 - 4*x^21 + 7*x^20 - 12*x^19 + 24*x^18 - 33*x^17 + 48*x^16 - 51*x^15 + 12*x^14 - 2*x^13 + 44*x^12 + 193*x^11 - 198*x^10 - 426*x^9 + 168*x^8 + 372*x^7 + 15*x^6 - 99*x^5 - 36*x^4 - 36*x^3 - 9*x^2 + 18*x + 9, 1)
 

Normalized defining polynomial

\( x^{22} - 4 x^{21} + 7 x^{20} - 12 x^{19} + 24 x^{18} - 33 x^{17} + 48 x^{16} - 51 x^{15} + 12 x^{14} - 2 x^{13} + 44 x^{12} + 193 x^{11} - 198 x^{10} - 426 x^{9} + 168 x^{8} + 372 x^{7} + 15 x^{6} - 99 x^{5} - 36 x^{4} - 36 x^{3} - 9 x^{2} + 18 x + 9 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-213248562554879935188951206691=-\,3^{20}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.53$
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:  $3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{15} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{19} + \frac{1}{3} a^{16} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7}$, $\frac{1}{3} a^{20} + \frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6}$, $\frac{1}{1634680822485814718037} a^{21} + \frac{68880981424053454730}{544893607495271572679} a^{20} - \frac{231850682655436553425}{1634680822485814718037} a^{19} - \frac{8539716408031046446}{52731639435026281227} a^{18} + \frac{16361554056604631831}{1634680822485814718037} a^{17} + \frac{124674935109529812650}{544893607495271572679} a^{16} + \frac{1324452661862312785}{12671944360355152853} a^{15} - \frac{206188410762782079463}{544893607495271572679} a^{14} + \frac{132441938264397809734}{544893607495271572679} a^{13} - \frac{224787062477973335174}{1634680822485814718037} a^{12} + \frac{93779495148690971960}{544893607495271572679} a^{11} - \frac{198530113710480754825}{1634680822485814718037} a^{10} + \frac{679505224285978641794}{1634680822485814718037} a^{9} - \frac{6734877894805756753}{1634680822485814718037} a^{8} + \frac{234168723596780972687}{544893607495271572679} a^{7} - \frac{160992432268013756717}{544893607495271572679} a^{6} - \frac{38329691857124877129}{544893607495271572679} a^{5} + \frac{36043082420196459274}{544893607495271572679} a^{4} - \frac{11110546745400298958}{544893607495271572679} a^{3} + \frac{124772143461366775164}{544893607495271572679} a^{2} + \frac{128821038783140053810}{544893607495271572679} a - \frac{83898051678234190428}{544893607495271572679}$

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:  $10$
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:  \( 1473183.5094 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_{11}$ (as 22T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 110
The 11 conjugacy class representatives for $F_{11}$
Character table for $F_{11}$

Intermediate fields

\(\Q(\sqrt{-11}) \), 11.1.139234453205859.1

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

Sibling fields

Degree 11 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.11.10.1$x^{11} - 3$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
3.11.10.1$x^{11} - 3$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$