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Properties

Label 22.0.11580323651...9227.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{11}\cdot 43^{2}\cdot 547^{2}\cdot 34374601^{2}$
Root discriminant $20.94$
Ramified primes $3, 43, 547, 34374601$
Class number $1$ (GRH)
Class group Trivial (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, -1, 6, -1, 20, 0, 38, 18, 40, 25, 5, 35, -23, 9, -15, -4, 3, -13, 10, -6, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 5*x^20 - 6*x^19 + 10*x^18 - 13*x^17 + 3*x^16 - 4*x^15 - 15*x^14 + 9*x^13 - 23*x^12 + 35*x^11 + 5*x^10 + 25*x^9 + 40*x^8 + 18*x^7 + 38*x^6 + 20*x^4 - x^3 + 6*x^2 - x + 1)
 
gp: K = bnfinit(x^22 - 2*x^21 + 5*x^20 - 6*x^19 + 10*x^18 - 13*x^17 + 3*x^16 - 4*x^15 - 15*x^14 + 9*x^13 - 23*x^12 + 35*x^11 + 5*x^10 + 25*x^9 + 40*x^8 + 18*x^7 + 38*x^6 + 20*x^4 - x^3 + 6*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} + 5 x^{20} - 6 x^{19} + 10 x^{18} - 13 x^{17} + 3 x^{16} - 4 x^{15} - 15 x^{14} + 9 x^{13} - 23 x^{12} + 35 x^{11} + 5 x^{10} + 25 x^{9} + 40 x^{8} + 18 x^{7} + 38 x^{6} + 20 x^{4} - x^{3} + 6 x^{2} - x + 1 \)

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:  \(-115803236519047451106947869227=-\,3^{11}\cdot 43^{2}\cdot 547^{2}\cdot 34374601^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.94$
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:  $3, 43, 547, 34374601$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{29275294621} a^{21} + \frac{4185761014}{29275294621} a^{20} + \frac{3928590006}{29275294621} a^{19} + \frac{13199236597}{29275294621} a^{18} - \frac{6415758335}{29275294621} a^{17} - \frac{7054629563}{29275294621} a^{16} + \frac{9524936744}{29275294621} a^{15} + \frac{11518178150}{29275294621} a^{14} + \frac{12952435522}{29275294621} a^{13} - \frac{5335611812}{29275294621} a^{12} + \frac{6660188026}{29275294621} a^{11} + \frac{11392218556}{29275294621} a^{10} + \frac{10656534332}{29275294621} a^{9} - \frac{2469442180}{29275294621} a^{8} - \frac{14191355256}{29275294621} a^{7} - \frac{6543744014}{29275294621} a^{6} + \frac{13864829787}{29275294621} a^{5} - \frac{13764315871}{29275294621} a^{4} + \frac{12328723558}{29275294621} a^{3} + \frac{8171879412}{29275294621} a^{2} - \frac{12442141888}{29275294621} a - \frac{7597210961}{29275294621}$

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:  \( -\frac{11307843807}{29275294621} a^{21} + \frac{18246944389}{29275294621} a^{20} - \frac{43384200134}{29275294621} a^{19} + \frac{38235754751}{29275294621} a^{18} - \frac{69224999870}{29275294621} a^{17} + \frac{84547144458}{29275294621} a^{16} + \frac{54054545054}{29275294621} a^{15} - \frac{11061081329}{29275294621} a^{14} + \frac{176701846103}{29275294621} a^{13} - \frac{39693760996}{29275294621} a^{12} + \frac{171776276246}{29275294621} a^{11} - \frac{266469120117}{29275294621} a^{10} - \frac{268235852086}{29275294621} a^{9} - \frac{178290882146}{29275294621} a^{8} - \frac{479126883259}{29275294621} a^{7} - \frac{312556665316}{29275294621} a^{6} - \frac{387667695384}{29275294621} a^{5} - \frac{111912562848}{29275294621} a^{4} - \frac{138072703532}{29275294621} a^{3} - \frac{107311486997}{29275294621} a^{2} - \frac{33643120766}{29275294621} a - \frac{293954466}{29275294621} \) (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:  \( 583781.984765 \) (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

\(\Q(\sqrt{-3}) \), 11.7.808524990121.1

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 $22$ R $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $18{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ $22$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
3Data not computed
$43$43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.7.0.1$x^{7} - 2 x + 9$$1$$7$$0$$C_7$$[\ ]^{7}$
43.7.0.1$x^{7} - 2 x + 9$$1$$7$$0$$C_7$$[\ ]^{7}$
547Data not computed
34374601Data not computed