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Properties

Label 22.10.4098058278...8729.7
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![-229, 1090, 12892, 1350, -37415, -14850, 9157, -7080, -3345, 5177, 10115, -3955, -106, -126, -145, -431, 31, 302, -157, 36, 2, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^21 + 2*x^20 + 36*x^19 - 157*x^18 + 302*x^17 + 31*x^16 - 431*x^15 - 145*x^14 - 126*x^13 - 106*x^12 - 3955*x^11 + 10115*x^10 + 5177*x^9 - 3345*x^8 - 7080*x^7 + 9157*x^6 - 14850*x^5 - 37415*x^4 + 1350*x^3 + 12892*x^2 + 1090*x - 229)
 
gp: K = bnfinit(x^22 - 5*x^21 + 2*x^20 + 36*x^19 - 157*x^18 + 302*x^17 + 31*x^16 - 431*x^15 - 145*x^14 - 126*x^13 - 106*x^12 - 3955*x^11 + 10115*x^10 + 5177*x^9 - 3345*x^8 - 7080*x^7 + 9157*x^6 - 14850*x^5 - 37415*x^4 + 1350*x^3 + 12892*x^2 + 1090*x - 229, 1)
 

Normalized defining polynomial

\( x^{22} - 5 x^{21} + 2 x^{20} + 36 x^{19} - 157 x^{18} + 302 x^{17} + 31 x^{16} - 431 x^{15} - 145 x^{14} - 126 x^{13} - 106 x^{12} - 3955 x^{11} + 10115 x^{10} + 5177 x^{9} - 3345 x^{8} - 7080 x^{7} + 9157 x^{6} - 14850 x^{5} - 37415 x^{4} + 1350 x^{3} + 12892 x^{2} + 1090 x - 229 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} + \frac{16}{47} a^{19} - \frac{3}{47} a^{18} + \frac{16}{47} a^{17} - \frac{20}{47} a^{16} + \frac{19}{47} a^{15} + \frac{12}{47} a^{14} + \frac{16}{47} a^{13} + \frac{11}{47} a^{11} - \frac{16}{47} a^{10} - \frac{5}{47} a^{9} + \frac{3}{47} a^{8} - \frac{11}{47} a^{7} + \frac{7}{47} a^{6} + \frac{14}{47} a^{5} + \frac{14}{47} a^{4} - \frac{13}{47} a^{3} - \frac{21}{47} a^{2} - \frac{16}{47} a - \frac{23}{47}$, $\frac{1}{130248142770186166791966360227064174724951754719789} a^{21} - \frac{367350108073168960956857890581176578244415174963}{130248142770186166791966360227064174724951754719789} a^{20} - \frac{39114683127559188771979747506997117764568510631090}{130248142770186166791966360227064174724951754719789} a^{19} + \frac{37889456281929706781659711069845342970600607193243}{130248142770186166791966360227064174724951754719789} a^{18} - \frac{30859047070714794694178018204434794153702555079303}{130248142770186166791966360227064174724951754719789} a^{17} + \frac{11164028632906766929488560896607829072156272661607}{130248142770186166791966360227064174724951754719789} a^{16} + \frac{56599083535376253352715793949364818253299248644821}{130248142770186166791966360227064174724951754719789} a^{15} + \frac{32617115782934872820009485828756851618581973856612}{130248142770186166791966360227064174724951754719789} a^{14} - \frac{14188397661385518167035811410112211703583227329464}{130248142770186166791966360227064174724951754719789} a^{13} - \frac{57537324272467589929988050188924661665303188489685}{130248142770186166791966360227064174724951754719789} a^{12} + \frac{43180090452767100792946662897533773780835402234006}{130248142770186166791966360227064174724951754719789} a^{11} + \frac{42500938654785834467348681409445684085288121005189}{130248142770186166791966360227064174724951754719789} a^{10} + \frac{25948100272826571769759156874515146225321090533031}{130248142770186166791966360227064174724951754719789} a^{9} + \frac{25758988242708493930085583700263636376778299879517}{130248142770186166791966360227064174724951754719789} a^{8} + \frac{20125282267720155252667167364878300618694931681867}{130248142770186166791966360227064174724951754719789} a^{7} + \frac{2759688219323079419530793842088195922386359964033}{130248142770186166791966360227064174724951754719789} a^{6} - \frac{40794474108229928167759545306221317911289159603541}{130248142770186166791966360227064174724951754719789} a^{5} + \frac{52468316006658326368406656881273242655266723170804}{130248142770186166791966360227064174724951754719789} a^{4} + \frac{54459649466027542146258016604764962526411492175395}{130248142770186166791966360227064174724951754719789} a^{3} + \frac{47675964302448849763081344267811607673980020978351}{130248142770186166791966360227064174724951754719789} a^{2} + \frac{8041043238337983491028863873615459750123264854480}{130248142770186166791966360227064174724951754719789} a - \frac{47115341907987959576923338941356423404611200334392}{130248142770186166791966360227064174724951754719789}$

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:  \( 355171257.143 \) (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