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Properties

Label 22.12.1781764468...3423.1
Degree $22$
Signature $[12, 5]$
Discriminant $-\,23^{20}\cdot 47^{3}$
Root discriminant $29.24$
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![-1, 7, 68, -21, -669, -55, 2661, -315, -4511, 2222, 1629, -1700, 1283, -441, -293, 293, -234, 79, -10, 1, 8, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 8*x^20 + x^19 - 10*x^18 + 79*x^17 - 234*x^16 + 293*x^15 - 293*x^14 - 441*x^13 + 1283*x^12 - 1700*x^11 + 1629*x^10 + 2222*x^9 - 4511*x^8 - 315*x^7 + 2661*x^6 - 55*x^5 - 669*x^4 - 21*x^3 + 68*x^2 + 7*x - 1)
 
gp: K = bnfinit(x^22 - x^21 + 8*x^20 + x^19 - 10*x^18 + 79*x^17 - 234*x^16 + 293*x^15 - 293*x^14 - 441*x^13 + 1283*x^12 - 1700*x^11 + 1629*x^10 + 2222*x^9 - 4511*x^8 - 315*x^7 + 2661*x^6 - 55*x^5 - 669*x^4 - 21*x^3 + 68*x^2 + 7*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 8 x^{20} + x^{19} - 10 x^{18} + 79 x^{17} - 234 x^{16} + 293 x^{15} - 293 x^{14} - 441 x^{13} + 1283 x^{12} - 1700 x^{11} + 1629 x^{10} + 2222 x^{9} - 4511 x^{8} - 315 x^{7} + 2661 x^{6} - 55 x^{5} - 669 x^{4} - 21 x^{3} + 68 x^{2} + 7 x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-178176446876650757881387571453423=-\,23^{20}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.24$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} - \frac{20}{47} a^{19} - \frac{1}{47} a^{18} - \frac{2}{47} a^{17} - \frac{6}{47} a^{16} - \frac{16}{47} a^{15} + \frac{7}{47} a^{14} - \frac{8}{47} a^{13} + \frac{3}{47} a^{12} - \frac{18}{47} a^{11} - \frac{12}{47} a^{10} - \frac{16}{47} a^{9} + \frac{21}{47} a^{8} + \frac{10}{47} a^{7} + \frac{8}{47} a^{6} + \frac{14}{47} a^{5} - \frac{12}{47} a^{4} - \frac{9}{47} a^{3} - \frac{13}{47} a^{2} + \frac{14}{47} a + \frac{18}{47}$, $\frac{1}{425898053226600832867712659} a^{21} + \frac{1989007593600691286645204}{425898053226600832867712659} a^{20} + \frac{117498688594178422693716686}{425898053226600832867712659} a^{19} + \frac{170578250013418609222668994}{425898053226600832867712659} a^{18} - \frac{144887019117811381430438995}{425898053226600832867712659} a^{17} + \frac{200481061476078625216497758}{425898053226600832867712659} a^{16} + \frac{16268887757075726206937202}{425898053226600832867712659} a^{15} + \frac{211477180491444648506237526}{425898053226600832867712659} a^{14} + \frac{209231164728626823228683577}{425898053226600832867712659} a^{13} - \frac{127130144962374466450614865}{425898053226600832867712659} a^{12} - \frac{9907190975584665200025121}{425898053226600832867712659} a^{11} - \frac{64712730362462966748049796}{425898053226600832867712659} a^{10} + \frac{201879247195669718534598846}{425898053226600832867712659} a^{9} + \frac{144060212589328123855406947}{425898053226600832867712659} a^{8} + \frac{766897244037297687200649}{425898053226600832867712659} a^{7} + \frac{1812470994946095783746095}{9061660706948953890802397} a^{6} + \frac{38386463051149692429821805}{425898053226600832867712659} a^{5} + \frac{27534034139214001710509574}{425898053226600832867712659} a^{4} - \frac{21151245889054238818044735}{425898053226600832867712659} a^{3} + \frac{46019853345228850178033522}{425898053226600832867712659} a^{2} - \frac{122817633417966463324369836}{425898053226600832867712659} a - \frac{77579966625758431812503264}{425898053226600832867712659}$

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:  $16$
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:  \( 106236735.142 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ R $22$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ $22$ R ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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