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Properties

Label 22.22.8813930939...3824.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{43}\cdot 7^{5}\cdot 11^{22}\cdot 137^{5}\cdot 1087^{5}$
Root discriminant $994.28$
Ramified primes $2, 7, 11, 137, 1087$
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![8, 2816, 123904, -7040, -619520, 4928, 1208064, -1408, -1208064, 176, 704704, -8, -256256, 0, 59840, 0, -8976, 0, 836, 0, -44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 44*x^20 + 836*x^18 - 8976*x^16 + 59840*x^14 - 256256*x^12 - 8*x^11 + 704704*x^10 + 176*x^9 - 1208064*x^8 - 1408*x^7 + 1208064*x^6 + 4928*x^5 - 619520*x^4 - 7040*x^3 + 123904*x^2 + 2816*x + 8)
 
gp: K = bnfinit(x^22 - 44*x^20 + 836*x^18 - 8976*x^16 + 59840*x^14 - 256256*x^12 - 8*x^11 + 704704*x^10 + 176*x^9 - 1208064*x^8 - 1408*x^7 + 1208064*x^6 + 4928*x^5 - 619520*x^4 - 7040*x^3 + 123904*x^2 + 2816*x + 8, 1)
 

Normalized defining polynomial

\( x^{22} - 44 x^{20} + 836 x^{18} - 8976 x^{16} + 59840 x^{14} - 256256 x^{12} - 8 x^{11} + 704704 x^{10} + 176 x^{9} - 1208064 x^{8} - 1408 x^{7} + 1208064 x^{6} + 4928 x^{5} - 619520 x^{4} - 7040 x^{3} + 123904 x^{2} + 2816 x + 8 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(881393093933942014560136225812507608113495413655720744560124493824=2^{43}\cdot 7^{5}\cdot 11^{22}\cdot 137^{5}\cdot 1087^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $994.28$
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:  $2, 7, 11, 137, 1087$
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 \(a\))

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

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:  $21$
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:  \( 243504752317000000000000000 \) (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

\(\Q(\sqrt{2}) \)

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 $20{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R R $20{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.0.1$x^{10} + 5 x^{2} - x + 5$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11Data not computed
$137$$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
137.10.5.2$x^{10} - 352275361 x^{2} + 1689160355995$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
137.10.0.1$x^{10} - x + 53$$1$$10$$0$$C_{10}$$[\ ]^{10}$
1087Data not computed