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Properties

Label 22.2.32217066567...0625.1
Degree $22$
Signature $[2, 10]$
Discriminant $5^{16}\cdot 11^{32}$
Root discriminant $105.46$
Ramified primes $5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T18

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1129, 1331, 121, -6655, -1210, 9317, 4719, -5324, -9438, 1331, 11011, -121, -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 - 121*x^11 + 11011*x^10 + 1331*x^9 - 9438*x^8 - 5324*x^7 + 4719*x^6 + 9317*x^5 - 1210*x^4 - 6655*x^3 + 121*x^2 + 1331*x + 1129)
 
gp: K = bnfinit(x^22 - 22*x^20 + 209*x^18 - 1122*x^16 + 3740*x^14 - 8008*x^12 - 121*x^11 + 11011*x^10 + 1331*x^9 - 9438*x^8 - 5324*x^7 + 4719*x^6 + 9317*x^5 - 1210*x^4 - 6655*x^3 + 121*x^2 + 1331*x + 1129, 1)
 

Normalized defining polynomial

\( x^{22} - 22 x^{20} + 209 x^{18} - 1122 x^{16} + 3740 x^{14} - 8008 x^{12} - 121 x^{11} + 11011 x^{10} + 1331 x^{9} - 9438 x^{8} - 5324 x^{7} + 4719 x^{6} + 9317 x^{5} - 1210 x^{4} - 6655 x^{3} + 121 x^{2} + 1331 x + 1129 \)

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:  \(322170665670052381217287483857714996337890625=5^{16}\cdot 11^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.46$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{45} a^{11} + \frac{4}{45} a^{9} - \frac{1}{45} a^{7} + \frac{13}{45} a^{5} + \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - \frac{11}{45} a - \frac{8}{45}$, $\frac{1}{45} a^{12} + \frac{4}{45} a^{10} - \frac{1}{45} a^{8} - \frac{2}{45} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} - \frac{11}{45} a^{2} + \frac{22}{45} a - \frac{1}{3}$, $\frac{1}{45} a^{13} - \frac{2}{45} a^{9} + \frac{2}{45} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} + \frac{22}{45} a^{2} + \frac{14}{45} a + \frac{2}{45}$, $\frac{1}{45} a^{14} - \frac{2}{45} a^{10} + \frac{2}{45} a^{8} + \frac{1}{15} a^{6} + \frac{1}{3} a^{5} + \frac{1}{5} a^{4} - \frac{8}{45} a^{3} + \frac{14}{45} a^{2} - \frac{13}{45} a - \frac{1}{3}$, $\frac{1}{45} a^{15} - \frac{1}{9} a^{9} + \frac{1}{45} a^{7} + \frac{1}{9} a^{5} - \frac{8}{45} a^{4} + \frac{4}{45} a^{3} + \frac{17}{45} a^{2} - \frac{7}{45} a - \frac{16}{45}$, $\frac{1}{45} a^{16} - \frac{1}{9} a^{10} + \frac{1}{45} a^{8} + \frac{1}{9} a^{6} - \frac{8}{45} a^{5} + \frac{4}{45} a^{4} + \frac{17}{45} a^{3} - \frac{7}{45} a^{2} - \frac{16}{45} a$, $\frac{1}{675} a^{17} - \frac{4}{675} a^{16} - \frac{2}{675} a^{15} + \frac{4}{675} a^{14} - \frac{1}{675} a^{13} + \frac{4}{675} a^{12} - \frac{7}{675} a^{11} + \frac{58}{675} a^{10} - \frac{17}{135} a^{9} - \frac{2}{45} a^{8} + \frac{2}{75} a^{7} - \frac{28}{225} a^{6} - \frac{22}{75} a^{5} + \frac{37}{75} a^{4} - \frac{103}{225} a^{3} + \frac{43}{675} a^{2} + \frac{182}{675} a - \frac{14}{675}$, $\frac{1}{675} a^{18} - \frac{1}{225} a^{16} - \frac{4}{675} a^{15} - \frac{2}{225} a^{12} + \frac{14}{225} a^{10} - \frac{8}{135} a^{9} - \frac{34}{225} a^{8} + \frac{2}{75} a^{7} - \frac{8}{225} a^{6} + \frac{52}{225} a^{5} + \frac{41}{225} a^{4} - \frac{218}{675} a^{3} + \frac{68}{225} a^{2} - \frac{77}{225} a - \frac{41}{675}$, $\frac{1}{675} a^{19} - \frac{1}{675} a^{16} - \frac{2}{225} a^{15} - \frac{1}{225} a^{14} + \frac{2}{225} a^{13} - \frac{1}{225} a^{12} + \frac{2}{225} a^{11} + \frac{29}{675} a^{10} + \frac{1}{225} a^{9} - \frac{8}{75} a^{8} + \frac{1}{9} a^{7} - \frac{4}{75} a^{6} - \frac{97}{225} a^{5} - \frac{119}{675} a^{4} + \frac{104}{225} a^{3} - \frac{49}{225} a^{2} - \frac{34}{135} a + \frac{4}{25}$, $\frac{1}{675} a^{20} + \frac{1}{135} a^{16} - \frac{1}{135} a^{15} - \frac{1}{135} a^{14} - \frac{4}{675} a^{13} - \frac{1}{135} a^{12} + \frac{7}{675} a^{11} - \frac{44}{675} a^{10} + \frac{8}{675} a^{9} + \frac{1}{15} a^{8} - \frac{1}{225} a^{7} - \frac{2}{15} a^{6} + \frac{43}{675} a^{5} - \frac{17}{45} a^{4} + \frac{73}{225} a^{3} - \frac{277}{675} a^{2} + \frac{61}{135} a - \frac{119}{675}$, $\frac{1}{675} a^{21} + \frac{1}{135} a^{15} + \frac{2}{225} a^{14} + \frac{2}{675} a^{12} + \frac{2}{225} a^{11} + \frac{2}{75} a^{10} + \frac{16}{135} a^{9} - \frac{16}{225} a^{8} + \frac{2}{45} a^{7} - \frac{2}{675} a^{6} - \frac{4}{9} a^{5} + \frac{88}{225} a^{4} + \frac{248}{675} a^{3} - \frac{1}{3} a^{2} + \frac{112}{225} a - \frac{2}{27}$

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:  \( 40057640608100 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T18:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2420
The 26 conjugacy class representatives for t22n18
Character table for t22n18 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 $20{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R $20{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R $20{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$11$11.11.16.4$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$[8/5]_{5}$
11.11.16.4$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$[8/5]_{5}$