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Properties

Label 22.10.4098058278...8729.5
Degree $22$
Signature $[10, 6]$
Discriminant $23^{21}\cdot 47^{3}$
Root discriminant $33.72$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T28

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![139, -800, 1290, -3601, 10009, -21117, 21927, 10505, -48721, 54245, -17678, -21230, 24766, -8995, -1068, 2710, -1469, 429, -17, -62, 35, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 + 35*x^20 - 62*x^19 - 17*x^18 + 429*x^17 - 1469*x^16 + 2710*x^15 - 1068*x^14 - 8995*x^13 + 24766*x^12 - 21230*x^11 - 17678*x^10 + 54245*x^9 - 48721*x^8 + 10505*x^7 + 21927*x^6 - 21117*x^5 + 10009*x^4 - 3601*x^3 + 1290*x^2 - 800*x + 139)
 
gp: K = bnfinit(x^22 - 9*x^21 + 35*x^20 - 62*x^19 - 17*x^18 + 429*x^17 - 1469*x^16 + 2710*x^15 - 1068*x^14 - 8995*x^13 + 24766*x^12 - 21230*x^11 - 17678*x^10 + 54245*x^9 - 48721*x^8 + 10505*x^7 + 21927*x^6 - 21117*x^5 + 10009*x^4 - 3601*x^3 + 1290*x^2 - 800*x + 139, 1)
 

Normalized defining polynomial

\( x^{22} - 9 x^{21} + 35 x^{20} - 62 x^{19} - 17 x^{18} + 429 x^{17} - 1469 x^{16} + 2710 x^{15} - 1068 x^{14} - 8995 x^{13} + 24766 x^{12} - 21230 x^{11} - 17678 x^{10} + 54245 x^{9} - 48721 x^{8} + 10505 x^{7} + 21927 x^{6} - 21117 x^{5} + 10009 x^{4} - 3601 x^{3} + 1290 x^{2} - 800 x + 139 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4098058278162967431271914143428729=23^{21}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.72$
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:  $23, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} - \frac{20}{47} a^{19} - \frac{22}{47} a^{18} - \frac{14}{47} a^{17} - \frac{20}{47} a^{16} + \frac{15}{47} a^{15} + \frac{5}{47} a^{14} + \frac{4}{47} a^{13} - \frac{6}{47} a^{12} + \frac{21}{47} a^{11} + \frac{18}{47} a^{10} + \frac{15}{47} a^{9} + \frac{13}{47} a^{8} - \frac{14}{47} a^{7} + \frac{2}{47} a^{6} - \frac{21}{47} a^{5} - \frac{16}{47} a^{4} + \frac{10}{47} a^{3} - \frac{4}{47} a^{2} + \frac{18}{47} a - \frac{9}{47}$, $\frac{1}{5168368839036024054340686659936298612726495319} a^{21} - \frac{45908703236069379512263982071770227054075638}{5168368839036024054340686659936298612726495319} a^{20} + \frac{1534812428336299422860993673177543890574771261}{5168368839036024054340686659936298612726495319} a^{19} + \frac{1452551632456926120909289122744011088509263044}{5168368839036024054340686659936298612726495319} a^{18} + \frac{1583573294455186754947376862074030765647155750}{5168368839036024054340686659936298612726495319} a^{17} + \frac{983529233319731709335172437537635319604027237}{5168368839036024054340686659936298612726495319} a^{16} + \frac{702395502445005912336229030684771893134337964}{5168368839036024054340686659936298612726495319} a^{15} - \frac{119721789439015870238671292102805508073089128}{5168368839036024054340686659936298612726495319} a^{14} + \frac{331470981393824987534557486623686917960945706}{5168368839036024054340686659936298612726495319} a^{13} + \frac{192390692003766994580561444199227192171571740}{5168368839036024054340686659936298612726495319} a^{12} + \frac{1938370212665383430763315474607470018424836413}{5168368839036024054340686659936298612726495319} a^{11} + \frac{1411251015537416340353956364937734966673185536}{5168368839036024054340686659936298612726495319} a^{10} + \frac{365756579723208901289295983437966360913206387}{5168368839036024054340686659936298612726495319} a^{9} + \frac{113751277887125992482981939788616656931849385}{5168368839036024054340686659936298612726495319} a^{8} - \frac{2125272214903891996820237549351355920875851028}{5168368839036024054340686659936298612726495319} a^{7} - \frac{1902506253996444976660392880146528480064773628}{5168368839036024054340686659936298612726495319} a^{6} + \frac{2462326957451932697697270288146261317256336917}{5168368839036024054340686659936298612726495319} a^{5} + \frac{440173843456493342327230291519011318450966594}{5168368839036024054340686659936298612726495319} a^{4} - \frac{35421365814505298534086887838903242709060768}{109965294447574979879589077870985076866521177} a^{3} - \frac{1388007821591276957060432660343130243155469528}{5168368839036024054340686659936298612726495319} a^{2} - \frac{2087153299002574697249349469363209183591316114}{5168368839036024054340686659936298612726495319} a - \frac{2152339721907680590831373737384767244949074370}{5168368839036024054340686659936298612726495319}$

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:  $15$
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:  \( 349912271.201 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T28:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 22528
The 208 conjugacy class representatives for t22n28 are not computed
Character table for t22n28 is not computed

Intermediate fields

\(\Q(\zeta_{23})^+\)

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ R $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
23Data not computed
47Data not computed