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Properties

Label 22.0.88663517253...1168.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{32}\cdot 11^{18}\cdot 13^{5}$
Root discriminant $34.92$
Ramified primes $2, 11, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T20

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10, 66, 121, -330, -1210, 462, 4719, -264, -9438, 66, 11011, -6, -8008, 0, 3740, 0, -1122, 0, 209, 0, -22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 22*x^20 + 209*x^18 - 1122*x^16 + 3740*x^14 - 8008*x^12 - 6*x^11 + 11011*x^10 + 66*x^9 - 9438*x^8 - 264*x^7 + 4719*x^6 + 462*x^5 - 1210*x^4 - 330*x^3 + 121*x^2 + 66*x + 10)
 
gp: K = bnfinit(x^22 - 22*x^20 + 209*x^18 - 1122*x^16 + 3740*x^14 - 8008*x^12 - 6*x^11 + 11011*x^10 + 66*x^9 - 9438*x^8 - 264*x^7 + 4719*x^6 + 462*x^5 - 1210*x^4 - 330*x^3 + 121*x^2 + 66*x + 10, 1)
 

Normalized defining polynomial

\( x^{22} - 22 x^{20} + 209 x^{18} - 1122 x^{16} + 3740 x^{14} - 8008 x^{12} - 6 x^{11} + 11011 x^{10} + 66 x^{9} - 9438 x^{8} - 264 x^{7} + 4719 x^{6} + 462 x^{5} - 1210 x^{4} - 330 x^{3} + 121 x^{2} + 66 x + 10 \)

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:  \(-8866351725365604006246875731591168=-\,2^{32}\cdot 11^{18}\cdot 13^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.92$
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:  $2, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{11} a^{20} - \frac{5}{11} a^{19} + \frac{4}{11} a^{18} - \frac{3}{11} a^{17} - \frac{3}{11} a^{16} + \frac{1}{11} a^{15} + \frac{3}{11} a^{14} - \frac{3}{11} a^{13} - \frac{4}{11} a^{12} - \frac{5}{11} a^{11} - \frac{1}{11} a^{10} + \frac{5}{11} a^{9} - \frac{4}{11} a^{8} + \frac{3}{11} a^{7} + \frac{3}{11} a^{6} - \frac{1}{11} a^{5} - \frac{3}{11} a^{4} + \frac{3}{11} a^{3} + \frac{4}{11} a^{2} + \frac{5}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{21} + \frac{1}{11} a^{19} - \frac{5}{11} a^{18} + \frac{4}{11} a^{17} - \frac{3}{11} a^{16} - \frac{3}{11} a^{15} + \frac{1}{11} a^{14} + \frac{3}{11} a^{13} - \frac{3}{11} a^{12} - \frac{4}{11} a^{11} - \frac{1}{11} a^{9} + \frac{5}{11} a^{8} - \frac{4}{11} a^{7} + \frac{3}{11} a^{6} + \frac{3}{11} a^{5} - \frac{1}{11} a^{4} - \frac{3}{11} a^{3} + \frac{3}{11} a^{2} + \frac{4}{11} a + \frac{5}{11}$

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:  \( a^{11} - 11 a^{9} + 44 a^{7} - 77 a^{5} + 55 a^{3} - 11 a - 3 \) (order $4$)
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:  \( 496691031.606 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T20:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4840
The 31 conjugacy class representatives for t22n20
Character table for t22n20 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \)

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

Sibling fields

Degree 22 sibling: 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ $22$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $22$ $20{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
2Data not computed
11Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.5.0.1$x^{5} - 2 x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
13.5.0.1$x^{5} - 2 x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
13.10.5.1$x^{10} - 676 x^{6} + 114244 x^{2} - 13366548$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$