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Properties

Label 22.0.10046547996...1443.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{11}\cdot 29^{2}\cdot 131^{2}\cdot 5399^{2}\cdot 367163^{2}$
Root discriminant $25.66$
Ramified primes $3, 29, 131, 5399, 367163$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 22T47

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 18, 12, 39, -20, 94, -72, 84, -148, 145, -129, 153, -96, 140, -41, 77, -18, 31, -4, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 7*x^20 - 4*x^19 + 31*x^18 - 18*x^17 + 77*x^16 - 41*x^15 + 140*x^14 - 96*x^13 + 153*x^12 - 129*x^11 + 145*x^10 - 148*x^9 + 84*x^8 - 72*x^7 + 94*x^6 - 20*x^5 + 39*x^4 + 12*x^3 + 18*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^22 - x^21 + 7*x^20 - 4*x^19 + 31*x^18 - 18*x^17 + 77*x^16 - 41*x^15 + 140*x^14 - 96*x^13 + 153*x^12 - 129*x^11 + 145*x^10 - 148*x^9 + 84*x^8 - 72*x^7 + 94*x^6 - 20*x^5 + 39*x^4 + 12*x^3 + 18*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 7 x^{20} - 4 x^{19} + 31 x^{18} - 18 x^{17} + 77 x^{16} - 41 x^{15} + 140 x^{14} - 96 x^{13} + 153 x^{12} - 129 x^{11} + 145 x^{10} - 148 x^{9} + 84 x^{8} - 72 x^{7} + 94 x^{6} - 20 x^{5} + 39 x^{4} + 12 x^{3} + 18 x^{2} + 4 x + 1 \)

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:  \(-10046547996724887091059294601443=-\,3^{11}\cdot 29^{2}\cdot 131^{2}\cdot 5399^{2}\cdot 367163^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.66$
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:  $3, 29, 131, 5399, 367163$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{13677239212089721} a^{21} + \frac{1293211240915228}{13677239212089721} a^{20} - \frac{5026308183262881}{13677239212089721} a^{19} + \frac{265569078960793}{13677239212089721} a^{18} + \frac{140683497308997}{13677239212089721} a^{17} + \frac{3092267571064062}{13677239212089721} a^{16} + \frac{704891312247647}{13677239212089721} a^{15} + \frac{3673414768924176}{13677239212089721} a^{14} - \frac{275290191621555}{13677239212089721} a^{13} + \frac{6536687791222354}{13677239212089721} a^{12} - \frac{2366917544829102}{13677239212089721} a^{11} - \frac{2404067940776095}{13677239212089721} a^{10} + \frac{3221173838721429}{13677239212089721} a^{9} - \frac{3768442768735910}{13677239212089721} a^{8} + \frac{5786275054318706}{13677239212089721} a^{7} - \frac{1418924701457527}{13677239212089721} a^{6} - \frac{259556439495174}{13677239212089721} a^{5} - \frac{1817689733634960}{13677239212089721} a^{4} + \frac{6618113169608472}{13677239212089721} a^{3} + \frac{2861743908395042}{13677239212089721} a^{2} - \frac{2205921994586860}{13677239212089721} a - \frac{4206275681874869}{13677239212089721}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{2685629293987735}{13677239212089721} a^{21} + \frac{2425506734097330}{13677239212089721} a^{20} - \frac{18387432369558813}{13677239212089721} a^{19} + \frac{9109022292712439}{13677239212089721} a^{18} - \frac{81671737681889821}{13677239212089721} a^{17} + \frac{42322341360437781}{13677239212089721} a^{16} - \frac{200264700221761619}{13677239212089721} a^{15} + \frac{99373435198554347}{13677239212089721} a^{14} - \frac{366524544292272972}{13677239212089721} a^{13} + \frac{247513061564379161}{13677239212089721} a^{12} - \frac{397088590715102046}{13677239212089721} a^{11} + \frac{353837120676200861}{13677239212089721} a^{10} - \frac{398897307145589961}{13677239212089721} a^{9} + \frac{417045500203026235}{13677239212089721} a^{8} - \frac{248328683492113525}{13677239212089721} a^{7} + \frac{224633125767905671}{13677239212089721} a^{6} - \frac{300744197938415810}{13677239212089721} a^{5} + \frac{63232402023086114}{13677239212089721} a^{4} - \frac{120862632834872891}{13677239212089721} a^{3} - \frac{7178304024482238}{13677239212089721} a^{2} - \frac{55662381431239864}{13677239212089721} a + \frac{1319886993338294}{13677239212089721} \) (order $6$)
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:  \( 2461752.74517 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T47:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 79833600
The 112 conjugacy class representatives for t22n47 are not computed
Character table for t22n47 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.9.7530807227563.1

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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ $18{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ $22$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$131$131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
131.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
131.10.0.1$x^{10} - x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
5399Data not computed
367163Data not computed